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6.2: Review of Power Series - Mathematics

6.2: Review of Power Series - Mathematics



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Many applications give rise to differential equations with solutions that can’t be expressed in terms of elementary functions such as polynomials, rational functions, exponential and logarithmic functions, and trigonometric functions. The solutions of some of the most important of these equations can be expressed in terms of power series. We’ll study such equations in this chapter. In this section we review relevant properties of power series. We’ll omit proofs, which can be found in any standard calculus text.

Definition (PageIndex{1})

An infinite series of the form

[label{eq:7.1.1} sum_{n=0}^infty a_n(x-x_0)^n,]

where (x_0) and (a_0), (a_1,) …, (a_n,) …are constants, is called a power series in (x-x_0.) We say that the power series Equation ef{eq:7.1.1} converges for a given (x) if the limit

[lim_{N oinfty} sum_{n=0}^Na_n(x-x_0)^n onumber]

exists(;) otherwise, we say that the power series diverges for the given (x.)

A power series in (x-x_0) must converge if (x=x_0), since the positive powers of (x-x_0) are all zero in this case. This may be the only value of (x) for which the power series converges. However, the next theorem shows that if the power series converges for some (x e x_0) then the set of all values of (x) for which it converges forms an interval.

Theorem (PageIndex{2})

For any power series

[sum_{n=0}^infty a_n(x-x_0)^n, onumber]

exactly one of the these statements is true(:)

  1. The power series converges only for (x=x_0.)
  2. The power series converges for all values of (x.)
  3. There’s a positive number (R) such that the power series converges if (|x-x_0|R).

In case (iii) we say that (R) is the radius of convergence of the power series. For convenience, we include the other two cases in this definition by defining (R=0) in case (i) and (R=infty) in case (ii). We define the open interval of convergence of (sum_{n=0}^infty a_n(x-x_0)^n) to be

[(x_{0}-R, x_{0}+R)quad ext{if}quad 0

If (R) is finite, no general statement can be made concerning convergence at the endpoints (x=x_0pm R) of the open interval of convergence; the series may converge at one or both points, or diverge at both.

Recall from calculus that a series of constants (sum_{n=0}^inftyalpha_n) is said to converge absolutely if the series of absolute values (sum_{n=0}^infty|alpha_n|) converges. It can be shown that a power series (sum_{n=0}^infty a_n(x-x_0)^n) with a positive radius of convergence (R) converges absolutely in its open interval of convergence; that is, the series

[sum_{n=0}^infty |a_n||x-x_0|^n onumber]

of absolute values converges if (|x-x_0|

The next theorem provides a useful method for determining the radius of convergence of a power series. It’s derived in calculus by applying the ratio test to the corresponding series of absolute values. For related theorems see Exercises 7.2.2 and 7.2.4.

Theorem (PageIndex{3})

Suppose there’s an integer (N) such that (a_n e0) if (nge N) and

[lim_{n oinfty}left|a_{n+1}over a_n ight|=L, onumber]

where (0le Lleinfty.) Then the radius of convergence of (sum_{n=0}^infty a_n(x-x_0)^n) is (R=1/ L,) which should be interpreted to mean that (R=0) if (L=infty,) or (R=infty) if (L=0).

Example (PageIndex{1})

Find the radius of convergence of the series:

  1. [sum_{n=0}^infty n!x^n onumber]
  2. [sum_{n=10}^infty (-1)^n {x^nover n!} onumber]
  3. [sum_{n=0}^infty 2^nn^2 (x-1)^n. onumber]

Solution a

Here (a_n=n!), so

[lim_{n oinfty}left|a_{n+1}over a_n ight|=lim_{n oinfty} {(n+1)!over n!}=lim_{n oinfty}(n+1)=infty. onumber]

Hence, (R=0).

Solution b

Here (a_n=(1)^n/n!) for (nge N=10), so

[lim_{n oinfty}left|a_{n+1}over a_n ight|=lim_{n oinfty} {n!over (n+1)!}=lim_{n oinfty}{1over n+1}=0. onumber]

Hence, (R=infty).

Solution c

Here (a_n=2^nn^2), so

[lim_{n oinfty}left|a_{n+1}over a_n ight|=lim_{n oinfty} {2^{n+1}(n+1)^2over2^nn^2}=2lim_{n oinfty}left(1+{1over n} ight)^2=2. onumber]

Hence, (R=1/2).

Taylor Series

If a function (f) has derivatives of all orders at a point (x=x_0), then the Taylor series of (f) about (x_0) is defined by

[sum_{n=0}^infty {f^{(n)}(x_0)over n!}(x-x_0)^n. onumber ]

In the special case where (x_0=0), this series is also called the Maclaurin series of (f).

Taylor series for most of the common elementary functions converge to the functions on their open intervals of convergence. For example, you are probably familiar with the following Maclaurin series:

[label{eq:7.1.2} e^{x} = sum_{n=0}^{infty} frac{x^{n}}{n!}, quad -infty

[label{eq:7.1.3} sin x = sum_{n=0}^{infty} (-1)^{n} frac{x^{2n+1}}{(2n+1)!}, quad -infty

[label{eq:7.1.4} cos x = sum_{n=0}^{infty} (-1)^{n} frac{x^{2n}}{(2n)!} quad -infty

[label{eq:7.1.5} frac{1}{1-x} = sum_{n=0}^{infty} x^{n} quad -1

Differentiation of Power Series

A power series with a positive radius of convergence defines a function

[f(x)=sum_{n=0}^infty a_n(x-x_0)^n onumber]

on its open interval of convergence. We say that the series represents (f) on the open interval of convergence. A function (f) represented by a power series may be a familiar elementary function as in Equations ef{eq:7.1.2} - ef{eq:7.1.5}; however, it often happens that (f) isn’t a familiar function, so the series actually defines (f).

The next theorem shows that a function represented by a power series has derivatives of all orders on the open interval of convergence of the power series, and provides power series representations of the derivatives.

Theorem (PageIndex{4}): A power series

A power series

[f(x)=sum_{n=0}^infty a_n(x-x_0)^n onumber]

with positive radius of convergence (R) has derivatives of all orders in its open interval of convergence, and successive derivatives can be obtained by repeatedly differentiating term by term(;) that is,

[egin{align} f'(x)&={sum_{n=1}^infty na_n(x-x_0)^{n-1}}label{eq:7.1.6}, f''(x)&={sum_{n=2}^infty n(n-1)a_n(x-x_0)^{n-2}},label{eq:7.1.7} &vdots& onumber f^{(k)}(x)&={sum_{n=k}^infty n(n-1)cdots(n-k+1)a_n(x-x_0)^{n-k}}label{eq:7.1.8}.end{align} onumber ]

Moreover, all of these series have the same radius of convergence (R.)

Example (PageIndex{2})

Let (f(x)=sin x). From Equation ef{eq:7.1.3},

[f(x)=sum_{n=0}^infty(-1)^n {x^{2n+1}over(2n+1)!}. onumber]

From Equation ef{eq:7.1.6},

[f'(x)=sum_{n=0}^infty(-1)^n{dover dx}left[x^{2n+1}over(2n+1)! ight]= sum_{n=0}^infty(-1)^n {x^{2n}over(2n)!}, onumber]

which is the series Equation ef{eq:7.1.4} for (cos x).

Uniqueness of Power Series

The next theorem shows that if (f) is defined by a power series in (x-x_0) with a positive radius of convergence, then the power series is the Taylor series of (f) about (x_0).

Theorem (PageIndex{5})

If the power series

[f(x)=sum_{n=0}^infty a_n(x-x_0)^n onumber]

has a positive radius of convergence, then

[label{eq:7.1.9} a_n={f^{(n)}(x_0)over n!};]

that is, (sum_{n=0}^infty a_n(x-x_0)^n) is the Taylor series of (f) about (x_0).

This result can be obtained by setting (x=x_0) in Equation ef{eq:7.1.8}, which yields

[f^{(k)}(x_0)=k(k-1)cdots1cdot a_k=k!a_k. onumber]

This implies that

[a_k={f^{(k)}(x_0)over k!}. onumber]

Except for notation, this is the same as Equation ef{eq:7.1.9}.

The next theorem lists two important properties of power series that follow from Theorem (PageIndex{4}).

Theorem (PageIndex{6})

  1. If [sum_{n=0}^infty a_n(x-x_0)^n=sum_{n=0}^infty b_n(x-x_0)^n onumber ] for all (x) in an open interval that contains (x_0,) then (a_n=b_n) for (n=0), (1), (2), ….
  2. If [sum_{n=0}^infty a_n(x-x_0)^n=0 onumber ] for all (x) in an open interval that contains (x_0,) then (a_n=0) for (n=0), (1), (2), ….

To obtain (a) we observe that the two series represent the same function (f) on the open interval; hence, Theorem (PageIndex{4}) implies that

[a_n=b_n={f^{(n)}(x_0)over n!},quad n=0,1,2, dots. onumber]

(b) can be obtained from (a) by taking (b_n=0) for (n=0), (1), (2), ….

Taylor Polynomials

If (f) has (N) derivatives at a point (x_0), we say that

[T_N(x)=sum_{n=0}^N{f^{(n)}(x_0)over n!}(x-x_0)^n onumber]

is the (N)-th Taylor polynomial of (f) about (x_0). This definition and Theorem (PageIndex{4}) imply that if

[f(x)=sum_{n=0}^infty a_n(x-x_0)^n, onumber]

where the power series has a positive radius of convergence, then the Taylor polynomials of (f) about (x_0) are given by

[T_N(x)=sum_{n=0}^N a_n(x-x_0)^n. onumber]

In numerical applications, we use the Taylor polynomials to approximate (f) on subintervals of the open interval of convergence of the power series. For example, Equation ef{eq:7.1.2} implies that the Taylor polynomial (T_N) of (f(x)=e^x) is

[T_N(x)=sum_{n=0}^N{x^nover n!}. onumber]

The solid curve in Figure (PageIndex{1}) is the graph of (y=e^x) on the interval ([0,5]). The dotted curves in Figure (PageIndex{1}) are the graphs of the Taylor polynomials (T_1), …, (T_6) of (y=e^x) about (x_0=0). From this figure, we conclude that the accuracy of the approximation of (y=e^x) by its Taylor polynomial (T_N) improves as (N) increases.

Shifting the Summation Index

In Definition (PageIndex{1}) of a power series in (x-x_0), the (n)-th term is a constant multiple of ((x-x_0)^n). This isn’t true in Equation ef{eq:7.1.6}, Equation ef{eq:7.1.7}, and Equation ef{eq:7.1.8}, where the general terms are constant multiples of ((x-x_0)^{n-1}), ((x-x_0)^{n-2}), and ((x-x_0)^{n-k}), respectively. However, these series can all be rewritten so that their (n)-th terms are constant multiples of ((x-x_0)^n). For example, letting (n=k+1) in the series in Equation ef{eq:7.1.6} yields

[label{eq:7.1.10} f'(x)=sum_{k=0}^infty (k+1)a_{k+1}(x-x_0)^k,]

where we start the new summation index (k) from zero so that the first term in Equation ef{eq:7.1.10} (obtained by setting (k=0)) is the same as the first term in Equation ef{eq:7.1.6} (obtained by setting (n=1)). However, the sum of a series is independent of the symbol used to denote the summation index, just as the value of a definite integral is independent of the symbol used to denote the variable of integration. Therefore we can replace (k) by (n) in Equation ef{eq:7.1.10} to obtain

[label{eq:7.1.11} f'(x)=sum_{n=0}^infty (n+1)a_{n+1}(x-x_0)^n,]

where the general term is a constant multiple of ((x-x_0)^n).

It isn’t really necessary to introduce the intermediate summation index (k). We can obtain Equation ef{eq:7.1.11} directly from Equation ef{eq:7.1.6} by replacing (n) by (n+1) in the general term of Equation ef{eq:7.1.6} and subtracting 1 from the lower limit of Equation ef{eq:7.1.6}. More generally, we use the following procedure for shifting indices.

Shifting the Summation Index in a Power Series

For any integer (k), the power series

[sum _ { n = n _ { 0 } } ^ { infty } b _ { n } left( x - x _ { 0 } ight) ^ { n - k } onumber ]

can be rewritten as

[sum _ { n = n _ { 0 } - k } ^ { infty } b _ { n + k } left( x - x _ { 0 } ight) ^ { n } onumber ]

that is, replacing (n) by (n + k) in the general term and subtracting k from the lower limit of summation leaves the series unchanged.

Example (PageIndex{3})

Rewrite the following power series from Equation ef{eq:7.1.7} and Equation ef{eq:7.1.8} so that the general term in each is a constant multiple of ((x-x_0)^n):

[(a) sum_{n=2}^infty n(n-1)a_n(x-x_0)^{n-2}quad (b) sum_{n=k}^infty n(n-1)cdots(n-k+1)a_n(x-x_0)^{n-k}. onumber ]

Solution a

Replacing (n) by (n+2) in the general term and subtracting (2) from the lower limit of summation yields

[sum_{n=2}^infty n(n-1)a_n(x-x_0)^{n-2}= sum_{n=0}^infty (n+2)(n+1)a_{n+2}(x-x_0)^n. onumber ]

Solution b

Replacing (n) by (n+k) in the general term and subtracting (k) from the lower limit of summation yields

[sum_{n=k}^infty n(n-1)cdots(n-k+1)a_n(x-x_0)^{n-k}= sum_{n=0}^infty (n+k)(n+k-1)cdots(n+1)a_{n+k}(x-x_0)^n. onumber ]

Example (PageIndex{4})

Given that

[f(x)=sum_{n=0}^infty a_nx^n, onumber]

write the function (xf'') as a power series in which the general term is a constant multiple of (x^n).

Solution

From Theorem (PageIndex{4}) with (x_0=0),

[f''(x)=sum_{n=2}^infty n(n-1)a_nx^{n-2}. onumber]

Therefore

[xf''(x)=sum_{n=2}^infty n(n-1)a_nx^{n-1}. onumber]

Replacing (n) by (n+1) in the general term and subtracting (1) from the lower limit of summation yields

[xf''(x)=sum_{n=1}^infty (n+1)na_{n+1}x^n. onumber]

We can also write this as

[xf''(x)=sum_{n=0}^infty (n+1)na_{n+1}x^n, onumber]

since the first term in this last series is zero. (We’ll see later that sometimes it is useful to include zero terms at the beginning of a series.)

Linear Combinations of Power Series

If a power series is multiplied by a constant, then the constant can be placed inside the summation; that is,

[csum_{n=0}^infty a_n(x-x_0)^n=sum_{n=0}^infty ca_n(x-x_0)^n. onumber]

Two power series

[f(x)=sum_{n=0}^infty a_n(x-x_0)^n quadmbox{ and }quad g(x)=sum_{n=0}^infty b_n(x-x_0)^n onumber]

with positive radii of convergence can be added term by term at points common to their open intervals of convergence; thus, if the first series converges for (|x-x_0|

[f(x)+g(x)=sum_{n=0}^infty(a_n+b_n)(x-x_0)^n onumber]

for (|x-x_0|

[c_1f(x)+c_2f(x)=sum_{n=0}^infty(c_1a_n+c_2b_n)(x-x_0)^n. onumber]

Example (PageIndex{5})

Find the Maclaurin series for (cosh x) as a linear combination of the Maclaurin series for (e^x) and (e^{-x}).

Solution

By definition,

[cosh x={1over2}e^x+{1over2}e^{-x}. onumber]

Since

[e^x=sum_{n=0}^infty {x^nover n!}quadmbox{ and }quad e^{-x}=sum_{n=0}^infty (-1)^n {x^nover n!}, onumber]

it follows that

[label{eq:7.1.12} cosh x=sum_{n=0}^infty {1over2}[1+(-1)^n]{x^nover n!}.]

Since

[{1over2}[1+(-1)^n]=left{egin{array}{cl}1&mbox{ if } n=2m,mbox{ an even integer}, 0&mbox{ if }n=2m+1,mbox{ an odd integer}, end{array} ight. onumber]

we can rewrite Equation ef{eq:7.1.12} more simply as

[cosh x=sum_{m=0}^infty{x^{2m}over(2m)!}. onumber]

This result is valid on ((-infty,infty)), since this is the open interval of convergence of the Maclaurin series for (e^x) and (e^{-x}).

Example (PageIndex{6})

Suppose

[y=sum_{n=0}^infty a_n x^n onumber]

on an open interval (I) that contains the origin.

  1. Express [(2-x)y''+2y onumber] as a power series in (x) on (I).
  2. Use the result of (a) to find necessary and sufficient conditions on the coefficients ({a_n}) for (y) to be a solution of the homogeneous equation

    [label{eq:7.1.13} (2-x)y''+2y=0]

    on (I).

Solution a

From Equation ef{eq:7.1.7} with (x_0=0),

[y''=sum_{n=2}^infty n(n-1)a_nx^{n-2}. onumber]

Therefore

[label{eq:7.1.14} egin{array}{rcl} (2-x)y''+2y &= 2y''-xy'+2y &= {sum_{n=2}^infty 2n(n-1)a_nx^{n-2} -sum_{n=2}^infty n(n-1)a_nx^{n-1} +sum_{n=0}^infty 2a_n x^n}. end{array}]

To combine the three series we shift indices in the first two to make their general terms constant multiples of (x^n); thus,

[label{eq:7.1.15} sum_{n=2}^infty 2n(n-1)a_nx^{n-2}=sum_{n=0}^infty2(n+2)(n+1)a_{n+2}x^n]

and

[label{eq:7.1.16} sum_{n=2}^infty n(n-1)a_nx^{n-1}=sum_{n=1}^infty(n+1)na_{n+1}x^n =sum_{n=0}^infty(n+1)na_{n+1}x^n,]

where we added a zero term in the last series so that when we substitute from Equation ef{eq:7.1.15} and Equation ef{eq:7.1.16} into Equation ef{eq:7.1.14} all three series will start with (n=0); thus,

[label{eq:7.1.17} (2-x)y''+2y=sum_{n=0}^infty [2(n+2)(n+1)a_{n+2}-(n+1)na_{n+1}+2a_n]x^n.]

Solution b

From Equation ef{eq:7.1.17} we see that (y) satisfies Equation ef{eq:7.1.13} on (I) if

[label{eq:7.1.18} 2(n+2)(n+1)a_{n+2}-(n+1)na_{n+1}+2a_n=0,quad n=0,1,2, dots.]

Conversely, Theorem (PageIndex{5})b implies that if (y=sum_{n=0}^infty a_nx^n) satisfies Equation ef{eq:7.1.13} on (I), then Equation ef{eq:7.1.18} holds.

Example (PageIndex{7})

Suppose

[y=sum_{n=0}^infty a_n (x-1)^n onumber]

on an open interval (I) that contains (x_0=1). Express the function

[label{eq:7.1.19} (1+x)y''+2(x-1)^2y'+3y]

as a power series in (x-1) on (I).

Solution

Since we want a power series in (x-1), we rewrite the coefficient of (y'') in Equation ef{eq:7.1.19} as (1+x=2+(x-1)), so Equation ef{eq:7.1.19} becomes

[2y''+(x-1)y''+2(x-1)^2y'+3y. onumber]

From Equation ef{eq:7.1.6} and Equation ef{eq:7.1.7} with (x_0=1),

[y'=sum_{n=1}^infty na_n(x-1)^{n-1}quadmbox{ and }quad y ''=sum_{n=2}^infty n(n-1)a_n(x-1)^{n-2}. onumber]

Therefore

[egin{aligned} 2y '' &= sum_{n=2}^infty 2n(n-1)a_n(x-1)^{n-2}, (x-1)y '' &= sum_{n=2}^infty n(n-1)a_n(x-1)^{n-1}, 2(x-1)^2y' &= sum_{n=1}^infty2na_n(x-1)^{n+1}, 3y &= sum_{n=0}^infty 3a_n (x-1)^n.end{aligned} onumber ]

Before adding these four series we shift indices in the first three so that their general terms become constant multiples of ((x-1)^n). This yields

[egin{align} 2y'' &= sum_{n=0}^infty 2(n+2)(n+1)a_{n+2}(x-1)^n,label{eq:7.1.20} (x-1)y'' &= sum_{n=0}^infty (n+1)na_{n+1}(x-1)^n, label{eq:7.1.21} 2(x-1)^2y' &= sum_{n=1}^infty 2(n-1)a_{n-1}(x-1)^n,label{eq:7.1.22} 3y &= sum_{n=0}^infty 3a_n (x-1)^n, label{eq:7.1.23}end{align}]

where we added initial zero terms to the series in Equation ef{eq:7.1.21} and Equation ef{eq:7.1.22}. Adding Equation ef{eq:7.1.20} –Equation ef{eq:7.1.23} yields

[egin{aligned} (1+x)y''+2(x-1)^2y'+3y &= 2y''+(x-1)y''+2(x-1)^2y'+3y &= sum_{n=0}^infty b_n (x-1)^n,end{aligned} onumber ]

where

[egin{align} b_0 &= 4a_2+3a_0, label{eq:7.1.24} b_n &= 2(n+2)(n+1)a_{n+2}+(n+1)na_{n+1}+2(n-1)a_{n-1}+3a_n,, nge1label{eq:7.1.25}.end{align}]

The formula Equation ef{eq:7.1.24} for (b_0) can’t be obtained by setting (n=0) in Equation ef{eq:7.1.25}, since the summation in Equation ef{eq:7.1.22} begins with (n=1), while those in Equation ef{eq:7.1.20}, Equation ef{eq:7.1.21}, and Equation ef{eq:7.1.23} begin with (n=0).


Assignments

There will be four exams and regular homework assignments. The first exam was Friday, June 5, 9 AM. The second was Wednesday, June 17. The third will be Tuesday, June 30 and the last will be Friday, July 10 (the last day of class).

Brief solutions to the penultimate problem set are available here.

The current (7/08) version of the problems for homework, beginning with question 88, is here (.pdf). The older problems are available here (.pdf) (numbers 1 to 49) and here (.pdf) (numbers 50 to 87).

  • Problem set 17 (to be turned in Friday, July 10, 12 PM): Questions 99-101. (There will be no additional problems assigned on Thursday, so you're welcome to hand this in Thursday to have it graded and returned on Friday.)
  • Problem set 16 (to be turned in Wednesday, July 8, 12:30 PM): From Monday: Questions 93-96.
    From Tuesday: Questions 97-8.
  • Problem set 15 (to be turned in Monday, July 6, 12:30 PM): There are 10 possible positions for the extra half hour of class: from 12 to 12:30 or from 2:30 to 3 on any day next week. Send me an e-mail stating those times with which you have a conflict, or that you have no conflict with any of those times. Then do questions 88-92.
  • Problem set 14 (to be turned in Thursday, July 2, 12:30 PM): Questions 84-87.
  • Exam 3: Questions and solutions in .pdf format.
  • Problem set 13 (to be turned in Monday, June 29, 12:30 PM): Questions 78-82. There is a typo in question 81: the words "below" and "above" should be switched.
  • Problem set 12 (to be turned in Friday, June 26, 12 PM): From Wednesday: Questions 71-75. Also re-do question 67 if you would like to submit it for credit.
    From Thursday: Questions 76, 77.
  • Problem set 11 (to be turned in Wednesday, June 24, 12:30 PM): From Monday: Questions 59, 60, 61, 62. From Tuesday: 65, 66, 67, 68. There is a typo in question 67. It should read, "what is the equation of the plane passing through (1, 2, 3) that cuts a tetrahedron from the first octant of minimal volume?" The first octant is the set of points in space such that x, y, z are all greater than or equal to 0. The 2-dimensional analogue of this problem would be, "What is the equation of the line passing through (1, 2) that cuts off a triangle from the first quadrant of minimal area?"
  • Problem set 10 (to be turned in Monday, June 22, 12:30 PM): Questions 56, 57, 58.
  • Problem set 9 (to be turned in Friday, June 19, 12 PM): Questions 50, 51, 52, 53, 55. Hint for 53: the direction vector of the line is parallel to both planes and so is perpendicular to the normal vectors of both planes. Hint for 55: ellipses are a lot like circles.
  • Problem set 8 (to be turned in Wednesday, June 17, 12:30 PM): Questions 48 and 49. For number 49, substitute a well-chosen value into one of the power series formulas you know in order to compute the series in question.
    Come prepared on Tuesday with any review questions for the upcoming exam.
  • Exam 2: Questions and solutions in .pdf format.
  • Problem set 7 (to be turned in Monday, June 15, 12:30 PM): Questions 42, 44, 45, 46, 47.
  • Problem set 6 (to be turned in Friday, June 12, 12:30 PM): From Wednesday: questions 36, 37.
    Added later Wednesday: 40, 41. For question 41, leave your answer in terms of an inverse trig function.
    Added Thursday: 38, 39, 43.
  • Problem set 5 (to be turned in Wednesday, June 10, 12:30 PM): From Monday: questions 19, 30, 32. (For number 19, the final answer should consists of the 18 functions in order using the symbols ">>," ">," "


To start analyzing this series, you could consider using Sterling's approximation $n!approx left(frac ne ight)^nsqrt <2pi n>$:

Of course this is only a lower limit to the approximated value, and that $sqrt n$ term still needs to be dealt with.

I had to do a lot of research on this, which was my benefit :)

Outline. Basically, I will use the Sophomore's Dream to deduce an integral expression for the answer.

Proof. We change variables writing $ ag<2a>x=expigl(-u/(n+1)igr)$ which lets us rewrite (1) as $int^<1>_<0>x^ln(x)^,mathrmx = (-1)^(n+1)^<-(n+1)>int^_<0>u^e^<-u>,mathrmu ag<2b>$ Observe the integral on the right hand side is precisely $n!$ (thanks to the Gamma function). And that concludes the proof for our lemma. QED.

Theorem. We claim $f(t) = 1+sum^_left(frac ight)^ = 1 + tint^<1>_<0>x^<-xt>,mathrmx. ag<3>$

Proof. We end up rewriting the integrand on the right hand side $x^ <-xt>= e^ <-xtln(x)>= sum^_frac<(-t)^>x^ln(x)^. ag<4a>$ We plug this back into the integral $int^<1>_<0>x^<-xt>,mathrmx= int^<1>_<0>sum^_frac<(-t)^>x^ln(x)^,mathrmx. ag<4b>$ Swap the sum and integral $int^<1>_<0>sum^_frac<(-t)^>x^ln(x)^,mathrmx =sum^_frac<(-t)^>int^<1>_<0>x^ln(x)^,mathrmx. ag<4c>$ We use our lemma to rewrite the right hand side as egin sum^_frac<(-t)^>int^<1>_<0>x^ln(x)^,mathrmx &= sum^_frac<(-t)^>left((-1)^(n+1)^<-n+1>n! ight) &= sum^_frac<>><(n+1)^>. ag <4c>end Then we just fiddle with arithmetic (multiply by $t$, and add 1) to get the series in question. That concludes our proof for the theorem. QED.

Remark. There is no closed form expression for the integral that I am aware of. Perhaps either the OP or some other user knows some fancy-pants way to evaluate the integral, but I do not know of one readily available :(


Power series

Our editors will review what you’ve submitted and determine whether to revise the article.

power series, in mathematics, an infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1 + x + x 2 + x 3 +⋯. Usually, a given power series will converge (that is, approach a finite sum) for all values of x within a certain interval around zero—in particular, whenever the absolute value of x is less than some positive number r, known as the radius of convergence. Outside of this interval the series diverges (is infinite), while the series may converge or diverge when x = ± r. The radius of convergence can often be determined by a version of the ratio test for power series: given a general power series a0 + a1x + a2x 2 +⋯, in which the coefficients are known, the radius of convergence is equal to the limit of the ratio of successive coefficients. Symbolically, the series will converge for all values of x such that

For instance, the infinite series 1 + x + x 2 + x 3 +⋯ has a radius of convergence of 1 (all the coefficients are 1)—that is, it converges for all −1 < x < 1—and within that interval the infinite series is equal to 1/(1 − x). Applying the ratio test to the series 1 + x/1! + x 2 /2! + x 3 /3! +⋯ (in which the factorial notation n! means the product of the counting numbers from 1 to n) gives a radius of convergence of so that the series converges for any value of x.

Most functions can be represented by a power series in some interval (see table ). Although a series may converge for all values of x, the convergence may be so slow for some values that using it to approximate a function will require calculating too many terms to make it useful. Instead of powers of x, sometimes a much faster convergence occurs for powers of (xc), where c is some value near the desired value of x. Power series have also been used for calculating constants such as π and the natural logarithm base e and for solving differential equations.

This article was most recently revised and updated by William L. Hosch, Associate Editor.


6.1 Power Series and Functions

A power series is a type of series with terms involving a variable. More specifically, if the variable is x, then all the terms of the series involve powers of x. As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions. In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power series.

Form of a Power Series

where x is a variable and the coefficients cn are constants, is known as a power series . The series

is an example of a power series. Since this series is a geometric series with ratio r = x , r = x , we know that it converges if | x | < 1 | x | < 1 and diverges if | x | ≥ 1 . | x | ≥ 1 .

Definition

is a power series centered at x = 0 . x = 0 . A series of the form

is a power series centered at x = a . x = a .

are both power series centered at x = 0 . x = 0 . The series

is a power series centered at x = 2 . x = 2 .

Convergence of a Power Series

Convergence of a Power Series

Proof

we conclude that, for all n ≥ N , n ≥ N ,

With this result, we can now prove the theorem. Consider the series

Definition

To determine the interval of convergence for a power series, we typically apply the ratio test. In Example 6.1, we show the three different possibilities illustrated in Figure 6.2.

Example 6.1

Finding the Interval and Radius of Convergence

For each of the following series, find the interval and radius of convergence.

Solution

Checkpoint 6.1

Find the interval and radius of convergence for the series ∑ n = 1 ∞ x n n . ∑ n = 1 ∞ x n n .

Representing Functions as Power Series

Being able to represent a function by an “infinite polynomial” is a powerful tool. Polynomial functions are the easiest functions to analyze, since they only involve the basic arithmetic operations of addition, subtraction, multiplication, and division. If we can represent a complicated function by an infinite polynomial, we can use the polynomial representation to differentiate or integrate it. In addition, we can use a truncated version of the polynomial expression to approximate values of the function. So, the question is, when can we represent a function by a power series?

Consider again the geometric series

Recall that the geometric series

As a result, we are able to represent the function f ( x ) = 1 1 − x f ( x ) = 1 1 − x by the power series

We now show graphically how this series provides a representation for the function f ( x ) = 1 1 − x f ( x ) = 1 1 − x by comparing the graph of f with the graphs of several of the partial sums of this infinite series.

Example 6.2

Graphing a Function and Partial Sums of its Power Series

Solution

Next we consider functions involving an expression similar to the sum of a geometric series and show how to represent these functions using power series.

Example 6.3

Representing a Function with a Power Series

Use a power series to represent each of the following functions f . f . Find the interval of convergence.

Solution

In the remaining sections of this chapter, we will show ways of deriving power series representations for many other functions, and how we can make use of these representations to evaluate, differentiate, and integrate various functions.

Section 6.1 Exercises

In the following exercises, state whether each statement is true, or give an example to show that it is false.

In the following exercises, find the radius of convergence R and interval of convergence for ∑ a n x n ∑ a n x n with the given coefficients a n . a n .

In the following exercises, find the radius of convergence of each series.

∑ k = 1 ∞ k ! 1 · 3 · 5 ⋯ ( 2 k − 1 ) x k ∑ k = 1 ∞ k ! 1 · 3 · 5 ⋯ ( 2 k − 1 ) x k

∑ k = 1 ∞ 2 · 4 · 6 ⋯ 2 k ( 2 k ) ! x k ∑ k = 1 ∞ 2 · 4 · 6 ⋯ 2 k ( 2 k ) ! x k

In the following exercises, use the ratio test to determine the radius of convergence of each series.

∑ n = 1 ∞ 2 3 n ( n ! ) 3 ( 3 n ) ! x n ∑ n = 1 ∞ 2 3 n ( n ! ) 3 ( 3 n ) ! x n

f ( x ) = x 2 1 − 4 x 2 a = 0 f ( x ) = x 2 1 − 4 x 2 a = 0

f ( x ) = x 2 5 − 4 x + x 2 a = 2 f ( x ) = x 2 5 − 4 x + x 2 a = 2

Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.

∑ k = 1 ∞ ( k − 1 2 k + 3 ) k x k ∑ k = 1 ∞ ( k − 1 2 k + 3 ) k x k

∑ k = 1 ∞ ( 2 k 2 − 1 k 2 + 3 ) k x k ∑ k = 1 ∞ ( 2 k 2 − 1 k 2 + 3 ) k x k

∑ n = 1 ∞ a n = ( n 1 / n − 1 ) n x n ∑ n = 1 ∞ a n = ( n 1 / n − 1 ) n x n

∑ n = 0 ∞ a 2 n x n ( H i n t : x = ± x 2 ) ∑ n = 0 ∞ a 2 n x n ( H i n t : x = ± x 2 )

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    Table of Contents

    Foreword
    Introduction and Contents
    Part One. General
    Chapter 1. Engineering Design and Mathematical Programming
    1.1. The Process of Engineering Design
    1.2. Application of Computers in System Design and Operation
    1.3. Methods of Optimization
    Chapter 2. An Outline of Power System Planning and Operation
    2.1. Objectives of System Planning
    2.2. Stages in System Planning and Design
    2.3. The Transition from Planning to Operation
    2.4. The Objectives of System Operation
    2.5. Stages in System Operation
    Chapter 3. Some Frequently Used Analytical Techniques
    3.1. Power Flows and Voltage
    3.2. The Nodal Impedance Matrix
    3.3. Fault Levels
    3.4. Transient and Steady-State Stability
    3.5. Some Useful Approximations
    3.6. System Costs
    Part Two. System Planning
    Chapter 4. The Estimation of Demand and Total Generation Requirement
    4.1. Estimation of Energy and Active Power Demands
    4.2. Estimation of Reactive Power
    4.3. The Estimation of Available Generation Capacity
    4.4. Reliability of Supply
    4.5. Gross Plant Margins and Standards of Supply in Practice
    Chapter 5. Standardization Studies for Network Plant
    5.1. Standardization Studies for One Stage of Development
    5.2. Standardization Studies for Several Stages of Development
    5.3. Fault Levels and Switchgear Rupturing Capacity
    Chapter 6. Generation Expansion Studies
    6.1. Comparative Economic Assessment of Individual Generation Projects
    6.2. The Investigation of Outline Generation Expansion Plans
    6.3. Simulation Models in Generation Expansion Planning
    6.4. A Heuristic Method to Investigate Outline Expansion Plans
    6.5. Linear Programming Models
    6.6. Dynamic Programming Formulations
    6.7. Other Non-linear Models
    6.8. Conclusion
    Chapter 7. Network Configuration Studies
    7.1. Typical Network Configurations
    7.2. The Configuration and Computation
    7.3. The Configuration Design Problem
    7.4. Costing of Schemes
    7.5. Security Criteria and Analysis of Network Viability
    7.6. Outline Design
    7.7. Configuration Design
    7.8. Configuration Synthesis Using Engineering Judgment
    7.9. Network Synthesis Using Mixed Engineering Judgment/Optimization Methods
    7.10. Configuration Synthesis Using Heuristic Logic
    7.11. Configuration Synthesis Using a Combinatorial Approach
    7.12. Two Recent Proposals for Configuration Synthesis
    7.13. Other Possible Approaches to Configuration Synthesis
    7.14. Conclusion
    Chapter 8. Probability and Planning
    8.1. Reliability Analysis in Network Planning
    8.2. Reliability Analysis on the Generation and Transmission System
    8.3. Risk and Uncertainty in Investment Decisions
    8.4. Conclusion
    Part Three. System Operation
    Chapter 9. Time Scales and Computation in System Operation
    Chapter 10. Load Prediction and Generation Capacity
    10.1. The Prediction of Demand
    10.2. Generation Capacity
    10.3. Optimum Maintenance Programming
    10.4. Fuel Supplies and Costs
    Chapter 11. Security Assessment
    11.1. Indications and Analysis of Insecure Operation
    11.2. Security Assessment against Excessive Current Flows
    11.3. Security Assessment against Excessive Fault Levels
    11.4. Security Assessment against Excessive Voltage Changes
    11.5. Security Assessment against System Instability
    11.6. The Present and Trends in Predictive Security Assessment
    11.7. The Present and Trends in On-line-Security Assessment
    Chapter 12. The Scheduling of Generating Plant
    12.1. A Manual Method of Scheduling
    12.2. Integer Programming Methods
    12.3. A Dynamic Programming Method
    12.4. Heuristic Methods
    12.5. Conclusion
    Chapter 13. The Dispatching of Generation
    13.1. Primary, Secondary and Tertiary Regulation
    13.2. Operation of Interconnected Areas
    13.3. Economic Dispatch Using the "Coordination Equations"
    13.4. Economic Dispatching Incorporating Group Transfer Analysis
    13.5. Economic Dispatching Incorporating Multiple-Load-Flow Analysis
    13.6. Dispatching Models Including an Exact Network Solution
    13.7. Transmission Loss Calculations and Optimum Dispatch
    13.8. System Control Centers
    13.9. Assessment of Optimum Network Configuration in Operation
    13.10. A Brief Note on the Operation of Hydrothermal Systems
    13.11. Computers and Dispatching in the Future
    Conclusion
    Appendix 1. Some Concepts in Probability Theory
    1.1. Markovian Systems
    1.2. Some Basic Equations in Reliability Theory
    1.3. Probability and the Binomial Distribution
    1.4. The Monte Carlo Method
    Appendix 2. Mathematical Programming
    2.1. Linear Programming
    2.2. Some Special Forms and Extensions of Linear Programming
    2.3. Non-linear Programming
    2.4. Dynamic Programming
    Appendix 3. Terms and Symbols Used
    3.1. Terms
    3.2. Symbols
    References
    Index
    Other Titles in the Series


    Exam 2

    • 2.5, 7.4: Differential Equations
    • 10.1-10.2: Improper Integrals
    • 11.1-11.4: Sequences and Series
    • 11.5-11.6: Power Series (you will only need be responsible for the information posted on the Power Series worksheets for Monday, 4/7, Wednesday, 4/9, and Friday, 4/11). In particular, you will NOT be tested on integration and differentiation of power series on Exam 2.
    • Chapter 11 Review Exercises: 1-8, 10, 13, 15, 16, 18, 19, 20, 21, 22, 37-48. Here are solutions for the Chapter 11 Review Exercises.

    Here are some practice exam questions for Exam 2. Note that inclusion or exclusion of a particular topic on the practice exam DOEST NOT mean that that topic will necessarily be included or excluded on the actual exam. The practice exam is just to give you some more practice problems to work on. You should, of course, study your class notes, homework problems, and quiz problems in addition to the practice exam. Here is an Answer Key for Practice Exam 2. Here are some worked out solutions and hints to the practice exam.

    Warning: Do not look at or print out the solutions to the above practice problems until after working on them yourself, taking some time, and going back later to try any problems you couldn't do the first time again. Doing the problems while looking at the answers renders them almost completely useless as preparation for taking an exam.


    Determining the Radius of Convergence of a Power Series

    We will now look at a technique for determining the radius of convergence of a power series using The Ratio Test for Positive Series

    Theorem 1: If $lim_ iggr vert frac<>> iggr vert = L$ where $L$ is a positive real number or $L = 0$ or $L = infty$ , then the power series $sum_^ a_n(x - c)^n$ has a radius of convergence $R = frac<1>$ where if $L = 0$ then $R = infty$ and if $L = infty$ then $R = 0$ .

    Let's now look at some examples of finding the radius of convergence of a power series.


    I plan to keep an up-to-date list of the topics, examples etc. covered in class. Unless stated otherwise, reference numbers refer to our textbook, W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (10th ed.), henceforth abbreviated as [BDP].

    Note: There is a free online access to WileyPLUS provided by the University on campus. This no-cost option is sufficient to complete the homework assignments for Math 201. However, it can be accessed only from these designated computer labs on campus, and it does not allow usage of any other online Wiley study tools.

    two attempts for every homework problem. Correct answers on second attempts are worth 80%.

    Three words about cheating:

    Midterm test

    The midterm test will be held on Saturday, October 24th, 2015 at 1:00 pm . You will write the midterm in N-RE 2-001 (our usual class room last names A - K) or N-RE 2-003 (last names L - Z).

    A midterm review session will be held for all sections on Thursday, 22 Oct, 5-7 pm in CCIS L2-190. Please make an effort to attend!
    The material for this review session can be found here.

    Need some practice material? The following is taken from last year's midterm: Midterm test 2014.

    Some of you have asked for additional practice material for homogeneous and Bernoulli equations. You may want to check out this for Bernoulli equations for homogeneous equations, try this and this (the latter also has some other substitutions).

    • Duration: 90 minutes.
    • Material covered: Up to, and including, power series solutions of linear differential equations, i.e., Chapters I to IV in class.
    • Some questions may be multiple choice.
    • NO calculators, formula sheets etc.!
    • NO cell-phones, i-pods, or other electronics!
    • Please bring a valid ID with you.
    • Good luck!

    The Math and Applied Sciences Centre is also offering a review session.

    Midterm test - Solutions

    Midterm test average:

    Final exam

    The final exam will be held on Saturday, December 19th, 2015 at 9:00 am in the Main Gym (Van Vliet building)

    The following rows have been reserved for you (section EB1):

    Please make sure you are seated in one of the correct rows.

    Some details concerning the final:

    • Duration: 120 minutes.
    • Material covered: Basically everything, but with a strong emphasis on the material covered in class after the midterm test.
    • A table of Laplace transform will be provided (from [BDP]).
    • NO calculators, formula sheets etc.!
    • NO cell-phones, i-pods, or other electronics!
    • Please bring a valid ID with you.
    • Good luck!

    A review session will be held for all sections on Wednesday, 9 Dec, 4:00-6:00pm in ETLC 1-003. Please make an effort to attend!
    The material for this review session can be found here and here.

    Other material

    Need help? The Decima Robinson Support Centre in CAB 528 offers free drop-in help sessions, Monday to Friday, 9:00 am to 3:00 pm. It's a great, friendly place, though quite busy at times.

    Your integration skills are a bit rusty? The Math and Applied Sciences Centre is running a Review of Integration Techniques.


    6.2: Review of Power Series - Mathematics

    Office hours: Tuesdays 5:00PM - 6:00PM and 8:30 - 9PM (Thursday office hours TBD), Hill 624 or by appointment.

    Email: cl.volkov at rutgers dot edu (for friends) / fq15 at scarletmail dot rutgers dot edu (for teaching)

    Lecture 2 (June 1, 2017). Lecture Notes, Workshop 1 (written by Dr. Scheffer), Writing Samples.
    The course materials mainly comes from Chapter 5 and 6 of Sundstrom's book.
    Also you can read Zorich's book, Section 1.2 and 1.3.
    All workshops are due 11:55PM the next Tuesday. So in case you have questions, you can discuss with me either before or after Tuesday's class.

    Lecture 3 (June 6, 2017). Lecture Notes
    For more details, please read Zorich, Section 2.1.
    An slightly different argument showing root 2 is not rational can be found in [Z], 2.2.2.c. The argument in the notes is modified from [A], Theorem 1.4.5.
    The construction of real numbers using Dedekind cuts can be found in [A], Section 8.6.

    Lecture 4 (June 8, 2017). Lecture Notes, Workshop 2
    Since I wasn't able to cover the density theorem, the workshop problem 5 is removed from this week's assignment.
    By now you should finish reading [A], Section 1.1 - 1.3 and Thompson-Brucker-Brucker, Elementary Real Analysis, Section 1.1 - 1.7.

    Lecture 5 (June 13, 2017). Lecture Notes
    It is very important that the Nested Interval Property applies only to closed intervals that are bounded. Think: which part of the proof fails when the intervals are not bounded.
    One can prove under the assumption of Archimedean Property, Nested Interval Property can imply Axiom of Completeness. Please see James Propp's paper Real Analysis in Reverse for more details. In the coming Chapter we will see a lot more such properties.

    Lecture 6 (June 15, 2017). Lecture Notes, Workshop 3
    In case you are interested in solving the optional workshop problem, please see the Notes on Countable Sets and Cantor's Diagonalization.
    The idea of Cantor's Diagonalization is to construct a decimal that is outside of the range of the function from the naturals to reals. Please see [A], Section 1.6 for details. In the note above you will find the most essential argument.
    By now you should finish reading Section 1.4 - 1.5 and 2.1 of the textbook, and Section 1.8 - 1.10, 2.1 - 2.4 of the TBB book
    About cardinalities, please read [Gamow] One Two Three Infinity, Chapter 1 and 2.

    Lecture 7 (June 20, 2017). Lecture Notes
    Here you should learn the technique of finding the N from the given conditions of convergence, instead of from the estimates.
    Also, to use the Algebraic Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.

    Lecture 8 (June 22, 2017). Lecture Notes, Workshop 4
    For the Order Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.
    Monotone Convergence gives a very convenient way of proving convergence, but usually does not tell you directly what the limit is. In general, getting the actual limit is usually difficult. In this class we only deal with some simple cases.
    Please make sure you can recall how to prove AoC implies MCT. Make a brief summary definitely helps.
    By now you should finish reading [A] 2.2 - 2.4, [TBB] 2.5 - 2.10.

    Lecture 9 (June 27, 2017). Lecture Notes
    In case you are struggling with the Workshop 4, Mr. Yang kindly wrote a guide to all the problems and agreed to share. Note that this is just a guide. The thinking process has been elaborated presented. Yet it does not make a proof. You still need to organize these thoughts into a proof.

    Lecture 10 (June 29, 2017). Lecture Notes, Workshop 5
    In case you are not satisfied with certain grade of the quizzes, or you have missed it due to any reason, please finish a write-up of the homework of the previous lecture and present your solution to me in person.
    For example, if you are not happy with your grades for Quiz 7, then you should do all the homework problems assigned in Lecture 7.
    I'll check a random problem to see if you really have good understanding towards it. If you have, then your quiz grade will be made to 8/10. To make up quizzes 1 - 9, your solutions must be presented before July 13th. After July 13th, the grades for Quiz 1 - 9 cannot be changed any more.
    By now you should finish reading [A] 2.5 - 2.6, [TBB] 2.11 - 2.12.

    Lecture 11 (July 4, 2017) No lectures today. Happy holiday!

    Lecture 12 (July 6, 2017). Midterm Exam, Workshop 6 (Written by Dr. Scheffer)
    Second chance policies: In case you didn't do well in the midterm, here is what you should do:

    • Study the course notes and other materials to make sure you know how to solve every problems in the exam.
    • Arrange a time for a Russian styled oral exam. I will pick a random problem in the exam.You will have 10 minutes for preparing the solutions. Then you should present the solution on the blackboard.
    • Books, pre-written notes are not allowed. The only thing you can refer to is the notes you generated in that 10 minutes.

    If your presentation is satisfactory, your midterm grade will be exonerated from the final grading computation. In other words, your grade will be computed as 60% Final + 20% Workshop + 10% Oral Quiz + 10% Written Quiz.

    Lecture 13 (July 11, 2017). Course Notes
    For those who missed tonight's lecture, please make sure you are capable of proving every single entry in the table on Page 9. In class I explained those examples on the blackboard. However the proof was only given orally. Please let me know if you are having trouble proving any items. I will be happy to supply an argument.
    The written quiz tonight is replaced as a Questionnaire regarding the midterm. Please find it in Sakai Assignments.

    Lecture 14 (July 13, 2017). Course Notes, Workshop 7
    Note: You don't need to worry the compactness part in either [A] or [TBB]. I did use the examples in [A] and the motivating comments in [TBB]. For Workshop 7, you don't need to know anything other than the currently posted course notes.
    By now you should finish reading [A] 3.2, [TBB] 4.1 - 4.4.

    Lecture 15 (July 18, 2017). Course Notes
    I have set up the system, so Workshop 6 can be (re)submitted until Aug. 4. Workshop 7 can be (re)submitted until July 25th.

    Lecture 16 (July 20, 2017). Course Notes, Workshop 8
    By now you should finish reading [A] 3.3, [TBB] 4.5 (Note that the Cousin's Property was not covered). You should start reading [A] 4.2 and [TBB] 5.1.
    Sorry for having delivered a stupidly organized lecture tonight. Hopefully the reorganized notes look better. Please let me know if you have troubles.

    Lecture 18 (July 27, 2017). Course Notes, Workshop 9
    By now you should finish reading [A] 4.1 - 4.3, [TBB] 5.1, 5.2, 5.4 and 5.5.
    On the second page of Workshop 9 you will find some comments to the exercises in [A]. Please at least attempt those problems I boldfaced.

    Lecture 19 (Aug. 1, 2017). Course Notes
    As we are about to finish Chapter 4 on Thursday, it is a very good point to review everything. If you have a good understanding on the materials in Chapter 1 to 4, you should feel no difficulty at all to understand Chapter 5, and most of the parts in Chapter 6 (until you arrive at the issue of uniform convergence of sequences and series of functions). If you are taking 312 next semester, your life will be easy for a while. So please do so without hesitation.
    For those who didn't do well in the quiz tonight, please answer the following questions:
            1. How many exercises did you attempt in 3.2, 3.3, 4.2, 4.3, 4.4?
            2. What kind of difficulty did you experience?
            3. Anything I can do to help?
    Please send your answers through emails. The grade for the quiz will be adjusted to 8/10 or your actual grade, whichever is higher.

    Lecture 20 (Aug. 3, 2017). Course Notes, Workshop 10
    Please attempt to prove those facts in Part 3 by yourself and do not read my argument unless you have no clue. My argument might be too complicated than it should be. The easiest way to simplify any complicated argument is to work your own argument without reading a word from the original one.
    The reason I chose these two easy problems for this last workshop assignment is to provide more free time for you to review the materials and attempt all the other problems in the book. Don't be lazy. You are not studying analysis for me, but to prepare for your future studies. The exercises in [A] is really the minimal amount you have to go through in order to master the skills.
    By now you should finish reading [A] 4.4 - 4.5 and [TBB] 5.6 - 5.9.

    Lecture 21 (Aug. 8, 2017). Course Notes.
    As you can see, if you have a solid understand for Chapter 1 through 4, there is no trouble for you to understand at least the theory of derivatives. The main challenge for this Chapter is how to use the results in real life. Please see Zorich's exercises for more practice.

    Lecture 22 (Aug. 10, 2017). Course Notes. Review of Chapter 1 to 4
    The exam will be held on next Tuesday. There will be 13 problems with 200 points. 150 points are considered as a perfect score. Please find more details on Sakai.
    By now you should finish reading [A] 5.1 - 5.3. If you have time, please also read [TBB] 7.1 - 7.7. We don't have enough time covering all these materials however the knowledge will be assumed in 312.

    In the Spring of 2017 I taught 640:244 (Differential Equation for Physics and Engineering) for Sections 20 - 22.
    I taught the same class in the past. Here are the materials I taught in Summer 2015. And here are the materials I used for teaching recitations of 244 in Spring 2015, Fall 2014, Spring 2014 and Fall 2013.

    Please find Dr. Shtelen's syllabus, schedule and homework assignments here.

    Please find the information concerning maple labs here.

    All announcements are to be posted on sakai. Please make sure that you have a working email address registered to the system.

    • MIT OCW Lectures on Differential Equations (Note that they have a different syllabus)
    • Dr. Z's Calc 4 Lecture Handouts (The mathematical central topic is covered and emphasized, with marginal topics discarded)
    • Maple Tutorial (Found and shared by Mr. Joshua Vigoureux).

    Week 2 (Jan. 25): Recitation Notes, Quiz 1.
    In case you have time, please also watch MIT Lecture 1 to further understand the geometric interpretation of ODE.
    Regarding the first order linear ODE, you can also check MIT Lecture 3 and read Dr. Z's notes for 2.1 for further understanding.
    Here are my own notes for Section 2.2 and 2.4

    Week 4 (Feb. 8): Recitation Notes (Part 1), Recitation Notes (Part 2) (allow me to reuse the notes in the past). Quiz 3
    In case you have time, please also watch MIT Lecture 5 and read Dr. Z's Notes on 2.5 (Note that Dr. Z used a different method).
    Here are my own notes for Section 2.7 and Section 3.1.

    Week 6 (Feb. 22): No recitation notes this week. Aside from those exam problems, I just went over the notes I announced in the previous week.
    The Quiz this week is take-home. Please carefully review Section 2.6 and 3.4.

    Week 7 (Mar. 1): Recitation Notes, Yet another take-home Quiz
    The principle I talked about in the recitation notes applies to Chapter 4 as well. You should keep in mind that
          1. First try templates, as well as exponential powers, are determined ONLY by the right hand side of the ODE.
          2. To determine how many times your template fails, you have to look at the characteristic roots, which are determined ONLY by the left hand side of the ODE.
    Please understand this set of recitation notes thoroughly.
    For 3.5 and 3.6, Dr. Z's notes may also be helpful: Notes on 3.5, Notes on 3.6
    My own notes on 3.5 (Part 1), 3.5 (Part 2), 3.5 (Part 3), 3.6, 3.4 and 3.7 (Course Plan), 3.4 and 3.7 (Notes Part 1), 3.7, 5.4 (Notes Part 2), 3.6, 3.8

    Week 8 (Mar. 8): Recitation Notes, Quiz 7
    Basically all the related materials were posted last week. So nothing more here.

    Week 9 (Mar. 15): Spring break. No recitation today. Enjoy!

    Week 10 (Mar. 22): Recitation Notes, Quiz 8, Quiz 8 Make-up
    Maple Lab 3 is due next week. Late submissions are allowed up to next Friday (Mar. 31, 2017).
    In case you have time, please read Dr. Z's notes on Section 4.1, Section 4.2, Section 4.3.
    My own notes on 4.1, 4.2, 4.3. Please find my notes on 3.8 above.

    Week 11 (Mar. 29): Recitation Notes for Linear Algebra, Quiz 9, Recitation Notes for 7.5, 7.6 and 7.8
    (Although these notes were written a while ago, it should be able to help)
    For the linear systems, Dr. Z's notes on 7.1, 7.4, 7.5, 7.6 and 7.8 should also be helpful.
    Please go over the (updated) Review Questions and make sure you are comfortable on everyone of it. I think it would help you better than any practice exam.

    Week 13 (Apr. 12): Quiz 11
    Aside from exam problems, all I talked about in class are in the recitation notes or previous week. Please go over it and especially make sure you know how to deal with complex eigenvalues.

    Week 14 (Apr. 19): Quiz 12
    Here are my summer course notes on Chapter 9: 9.1, 9.2, 9.3, 9.4, 9.4 leftovers (Shared by Ms. Shawnie Caslin). Also please watch MIT Lecture 31 for how to deal with nonlinear systems.
    I wasn't able to type up the notes for finding global trajectories. In case you have taken neat notes, please don't hesitate to share.
    Maple Lab 5 is supposed to due yesterday. Late submissions are accepted until next Tuesday (Apr. 25).

    For 244 students, I have two requirements

    If you don't know how to manipulate logarithm, please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch10_SE.pdf
    Please read Section 10.5 on page 45 in the pdf file (page 733 in the book), try all example problems, and do Exercise 44 - 61 on page 51 in the pdf file (Page 740 in the book).

    If you are not very fluent with the quadratic equations (e.g. always use the root formula), please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch08_SE.pdf
    Read Section 8.1, 8.2 , try all example problems, and do Exercise 66 - 83 on page 23 in the pdf file (Page 573 in the book). Make sure you understand all the related methods

    In particular, if you have never seen criss-cross factorization before, please check the youtube videos
    Criss-Cross Method 1, Criss-Cross Method 2, Criss-Cross Method 3 and Criss-Cross Method 4.

    If you have never seen matrices before, please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch03_SE.pdf
    Read Section 3.6 , try all example problems, and do Exercise 15 - 23, 46 - 49 on page 51 - 52 in the pdf file (page 227 - 228 in the book).
    Read Section 3.7 , try all example problems, and do Exercise 2 - 7, 20 - 25, 35 - 40 on page 63 - 64 in the pdf file (page 239 - 240 in the book).
    After you work on this topic, try the problems of the attendence quiz at Lecture 15 and you will find it easy to play.

    If you keep on making mistakes on exponentials, please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch01_SE.pdf
    Read Section 1.8 , try all example problems, and do Exercise 59 - 84 on page 88 in the pdf file (page 88 in the book).

    If you don't know how to divide a polynomial, please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch05_SE.pdf
    Read Section 5.3 , try all example problems, and do Exercise 27 - 42 on page 31 in the pdf file (page 339 in the book).
    After you have done the work, please compare to the technique I used on dealing with t/(t+1) or -2-t/(t+1) in class. You will see that this is actually the simplest example of division.

    If you are not fluent on simplifications of rational functions, please find
    http://people.ucsc.edu/

    miglior/chapter%20pdf/Ch06_SE.pdf
    Read Section 6.1 - 6.4 , try all example problems, and do Exercise 29 - 48 on page 61 - 62 in the pdf file (page 463 - 464 in the book).

    If you are not fluent on playing with trigonometric functions, please find
    http://www.eht.k12.nj.us/

    staffoch/Textbook/chapter04.pdf
    Read Section 4.3 , make sure you memorize the table of the values of sine, cosine and tangent on usual special angles on page 23 of the PDF file (page 279 in the book)
    and do Exercise 17 - 26 on page 28 of the pdf file (page 284 in the book)
    Read Section 4.5 , make sure you can recognize, distinguish different graphs of the trignometric functions and manipulate them by scaling and translation , and do Exercise 3 - 14, 23 - 16 on page 48 in the pdf file (page 304 in the book)

    If you are not fluent on factorizing polynomials, please find
    http://people.ucsc.edu/

    Please make sure you have a solid understanding on the math 300 class (Introduction to Mathematical Reasoning). You can review the knowledge using the following material
    Dr. Sussmann's notes on Math 300, Lecture 2, 3 and 4
    This set of notes summarizes the most essential knowledge in that class. On his course website you'll find more related material for reviewing.

    Please recall the knowledge of Calculus I, especially the graphs of the most commonly seen elementary functions. You can check the following file to recall the knowledge:
    Table of Common Graphs
    Although the main focus is to formulate rigorous argument, in many cases this process is facilitated by the intuition from the graphs.
    Also I'll assume a solid basis of computational skills for this class. Please try problems in Chapter 1 and 2 of famous Russian book
    3193 Problems in Mathematical Analysis
    to test your skills.