# 6.1: Review of Power Series - Mathematics

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Before we go on to solving differential equations using power series, it would behoove you to go back to you calculus notes and review power series. There is one topic that was a small detail in first year calculus, but will be a main issue for solving differential equations. This is the technique of changing the index.

Example (PageIndex{1})

Change the index and combine the power series

[ sum_{n=1}^infty n,a_n,x^{n+1} + sum_{n=0}^infty a_n,x^n. onumber ]

Solution

There are two issues here: The first is the the powers of (x) are different and the second is that the summations begin at different values. To make the powers of (x) the same we perform the substitution

[u = n + 1, ;;; n = u - 1. onumber]

Notice that when (n = 1), (u = 2) and when (n) is infinity so is (u). We can write

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

Since (u) is a dummy index, we can rename it (n) to get

[ sum_{u=2}^{infty} (u-1),a_{u-1},x^u = sum_{n=2}^{infty} (n-1),a_{n-1},x^n. onumber]

We now need to find

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

The trouble now is that the starting numbers are different for the two series. We can pull the first two terms out of the second series to get

[ sum_{n=0}^{infty} a_{n},x^n = a_0 + a_1,x + sum_{n=2}^{infty} a_n, x^n onumber ]

putting this together we get

[egin{align*} sum_{n=0}^{infty} (n-1), a_{n-1},x^n + a_0+a_1,x+ sum_{n=2}^{infty} a_n, x^n [4pt] &= a_0+a_1,x+ sum_{n=2}^{infty} left[ (n-1), a_{n-1} +a_n ight]x^n. end{align*} ]

## Power series

anx n is called the n th term of the power series.

an is called the n th coefficient of the power series.

Notice that we are adding up terms with increasing powers of (x - c), hence the name power series.

Power series are used to approximate functions that are difficult to calculate exactly, such as tan -1 (x) and sin(x), using an infinite series of polynomials. Power series are often used to approximate important quantities and functions such as π, e , and , an important function in statistics.

In real life, we cannot add an infinite number of terms together since any computer can only hold a certain amount of memory. Therefore, we approximate a power series using the th partial sum of a power series, denoted Sn(x). To produce the n th partial sum, we cut off the infinite series after the n th term, getting rid of all terms with powers of (x - c) higher than n. We only keep the first n + 1 terms of the power series (remember that we start from the 0 th term which is f(c)). The n th partial sum is defined as:

exists, then the power series is said to converag eat x0 . Otherwise, the power series diverges at x0.

In cases where c = 0, the infinite sum is

The domain of f, often called the interval of convergence (IOC), is the set of all x-values such that the power series converges.

• For a given power series, it can be proven that either the IOC = (-∞,∞), meaning that the series converges for all x, or there exists a finite non-negative number R ≥ 0, called the radius of convergence (ROC), such that the series converges whenever |x - c| R.
• By convention, when the IOC of f is (-∞, ∞), the ROC is R = +∞.
• When R = 0, the IOC consists only of c.
• We can find the IOC by first finding the ROC with the ratio or root test, and then testing the endpoints c &pm R with some other test like integral, comparison, alternating series, p-series, etc.
• If the ROC is a finite number R, the IOC will always be of one of the following forms:
• [c - R, c + R]
• (c - R, c + R]
• [c - R, c + R)
• (c - R, c + R)

### Geometric series

A geometric series is one of the most important examples of a power series. In a geometric series, all the a's are the same and c = 0:

The n th partial sum of f is given by:

Sn(x) = a + ax + ax 2 + ax 3 + ax 4 + . + ax n

This can be checked by multiplying both expressions for Sn(x) by x - 1 and realizing that all powers of x cancel out except the 0 th and n+1 th :

Sn(x) alternates between a and 0 so does not exist and the series diverges.

if and only if |x| . Therefore, the geometric series has ROC 1 centered at 0 and IOC = (-1, 1).

Note that these questions are the same as asking whether non-zero $a_i$, $b_i$ can satisfy $sum_^infty a_kx^=0$ for all $x$ (or for $x>1$). This assumes the series converge absolutely, so that we can combine them. If we're not assuming absolute convergence, then can prove absolute convergence assuming that the $b_i$ are positive (this can be relaxed a bit, but not done away with).

Assuming the $b_i$ are strictly decreasing, we can answer it as follows. Assume the series converges absolutely for $xge A$ (that such an $A> 1$ exists will be shown at the end). Without loss of generality, assume $a_1 e 0$. For $xge A$, we have $0=a_1 x^+sum_^infty a_nx^$ So we solve for $a_1$ and take absolute values $|a_1|le x^<-b_1>sum_^infty |a_n|x^=x^<-b_1+b_2>sum_^infty |a_n|x^$ Since the $b_i$ are strictly decreasing, both $-b_1+b_2$ and $b_n-b_2$ are negative (or zero, for $n=2$), thus for $xge A$, $x^le A^$. Therefore, for $xge A$, $|a_1|le x^<-b_1+b_2>sum_^infty |a_n|A^$ Since $x$ can be arbitrarily large, this implies that $a_1=0$. Contradiction.

We now prove the existence of such an $A$. Assume the series converges at $x_0$. Fix $epsilon>0$ and set $x_n=x_0e^)/b_n>$ Then $|a_n|x_n^=|a_n|x_0^(x_n/x_0)^le (|a_n|x_0^)n^<-(1+epsilon)>$ Set $A=limsup_n x_n$. Since the series converges at $x_0$, the terms $|a_n|x_0^$ are bounded, hence the series will converge absolutely for $xge A$. A priori, $A$ could be infinite. But this won't happen as the $b_n$ are decreasing and positive.

## 6.1: Review of Power Series - Mathematics

We’ve spent quite a bit of time talking about series now and with only a couple of exceptions we’ve spent most of that time talking about how to determine if a series will converge or not. It’s now time to start looking at some specific kinds of series and we’ll eventually reach the point where we can talk about a couple of applications of series.

In this section we are going to start talking about power series. A power series about a, or just power series, is any series that can be written in the form,

where (a) and () are numbers. The ()’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of (x). That is different from any other kind of series that we’ve looked at to this point. In all the prior sections we’ve only allowed numbers in the series and now we are allowing variables to be in the series as well. This will not change how things work however. Everything that we know about series still holds.

In the discussion of power series convergence is still a major question that we’ll be dealing with. The difference is that the convergence of the series will now depend upon the values of (x) that we put into the series. A power series may converge for some values of (x) and not for other values of (x).

Before we get too far into power series there is some terminology that we need to get out of the way.

First, as we will see in our examples, we will be able to show that there is a number (R) so that the power series will converge for, (left| ight| < R) and will diverge for (left| ight| > R). This number is called the radius of convergence for the series. Note that the series may or may not converge if (left| ight| = R). What happens at these points will not change the radius of convergence.

Secondly, the interval of all (x)’s, including the endpoints if need be, for which the power series converges is called the interval of convergence of the series.

These two concepts are fairly closely tied together. If we know that the radius of convergence of a power series is (R) then we have the following.

The interval of convergence must then contain the interval (a - R < x < a + R) since we know that the power series will converge for these values. We also know that the interval of convergence can’t contain (x)’s in the ranges (x < a - R) and (x > a + R) since we know the power series diverges for these value of (x). Therefore, to completely identify the interval of convergence all that we have to do is determine if the power series will converge for (x = a - R) or (x = a + R). If the power series converges for one or both of these values then we’ll need to include those in the interval of convergence.

Before getting into some examples let’s take a quick look at the convergence of a power series for the case of (x = a). In this case the power series becomes,

and so the power series converges. Note that we had to strip out the first term since it was the only non-zero term in the series.

It is important to note that no matter what else is happening in the power series we are guaranteed to get convergence for (x = a). The series may not converge for any other value of (x), but it will always converge for (x = a).

Let’s work some examples. We’ll put quite a bit of detail into the first example and then not put quite as much detail in the remaining examples.

Okay, we know that this power series will converge for (x = - 3), but that’s it at this point. To determine the remainder of the (x)’s for which we’ll get convergence we can use any of the tests that we’ve discussed to this point. After application of the test that we choose to work with we will arrive at condition(s) on (x) that we can use to determine which values of (x) for which the power series will converge and which values of (x) for which the power series will diverge. From this we can get the radius of convergence and most of the interval of convergence (with the possible exception of the endpoints).

With all that said, the best tests to use here are almost always the ratio or root test. Most of the power series that we’ll be looking at are set up for one or the other. In this case we’ll use the ratio test.

Before going any farther with the limit let’s notice that since (x) is not dependent on the limit it can be factored out of the limit. Notice as well that in doing this we’ll need to keep the absolute value bars on it since we need to make sure everything stays positive and (x) could well be a value that will make things negative. The limit is then,

So, the ratio test tells us that if (L < 1) the series will converge, if (L > 1) the series will diverge, and if (L = 1) we don’t know what will happen. So, we have,

We’ll deal with the (L = 1) case in a bit. Notice that we now have the radius of convergence for this power series. These are exactly the conditions required for the radius of convergence. The radius of convergence for this power series is (R = 4).

Now, let’s get the interval of convergence. We’ll get most (if not all) of the interval by solving the first inequality from above.

So, most of the interval of validity is given by ( - 7 < x < 1). All we need to do is determine if the power series will converge or diverge at the endpoints of this interval. Note that these values of (x) will correspond to the value of (x) that will give (L = 1).

The way to determine convergence at these points is to simply plug them into the original power series and see if the series converges or diverges using any test necessary.

(x = - 7):
In this case the series is,

This series is divergent by the Divergence Test since (mathop limits_ n = infty e 0).

(x = 1):
In this case the series is,

This series is also divergent by the Divergence Test since (mathop limits_ ight)^n>n) doesn’t exist.

So, in this case the power series will not converge for either endpoint. The interval of convergence is then,

In the previous example the power series didn’t converge for either endpoint of the interval. Sometimes that will happen, but don’t always expect that to happen. The power series could converge at either both of the endpoints or only one of the endpoints.

Let’s jump right into the ratio test.

So we will get the following convergence/divergence information from this.

We need to be careful here in determining the interval of convergence. The interval of convergence requires (left| ight| < R) and (left| ight| > R). In other words, we need to factor a 4 out of the absolute value bars in order to get the correct radius of convergence. Doing this gives,

So, the radius of convergence for this power series is (R = frac<1><8>).

Now, let’s find the interval of convergence. Again, we’ll first solve the inequality that gives convergence above.

(displaystyle x = frac<<15>><8>):
The series here is,

This is the alternating harmonic series and we know that it converges.

(displaystyle x = frac<<17>><8>):
The series here is,

This is the harmonic series and we know that it diverges.

So, the power series converges for one of the endpoints, but not the other. This will often happen so don’t get excited about it when it does. The interval of convergence for this power series is then,

We now need to take a look at a couple of special cases with radius and intervals of convergence.

We’ll start this example with the ratio test as we have for the previous ones.

At this point we need to be careful. The limit is infinite, but there is that term with the (x)’s in front of the limit. We’ll have (L = infty > 1) provided (x e - frac<1><2>).

So, this power series will only converge if (x = - frac<1><2>). If you think about it we actually already knew that however. From our initial discussion we know that every power series will converge for (x = a) and in this case (a = - frac<1><2>). Remember that we get (a) from ( ight)^n>), and notice the coefficient of the (x) must be a one!

In this case we say the radius of convergence is (R = 0) and the interval of convergence is (x = - frac<1><2>), and yes we really did mean interval of convergence even though it’s only a point.

In this example the root test seems more appropriate. So,

So, since (L = 0 < 1) regardless of the value of (x) this power series will converge for every (x).

In these cases, we say that the radius of convergence is (R = infty ) and interval of convergence is ( - infty < x < infty ).

So, let’s summarize the last two examples. If the power series only converges for (x = a) then the radius of convergence is (R = 0) and the interval of convergence is (x = a). Likewise, if the power series converges for every (x) the radius of convergence is (R = infty ) and interval of convergence is ( - infty < x < infty ).

Let’s work one more example.

First notice that (a = 0) in this problem. That’s not really important to the problem, but it’s worth pointing out so people don’t get excited about it.

The important difference in this problem is the exponent on the (x). In this case it is 2(n) rather than the standard (n). As we will see some power series will have exponents other than an (n) and so we still need to be able to deal with these kinds of problems.

This one seems set up for the root test again so let’s use that.

So, we will get convergence if

The radius of convergence is NOT 3 however. The radius of convergence requires an exponent of 1 on the (x). Therefore,

[eginsqrt <> & < sqrt 3 left| x ight| & < sqrt 3 end]

Be careful with the absolute value bars! In this case it looks like the radius of convergence is (R = sqrt 3 ). Notice that we didn’t bother to put down the inequality for divergence this time. The inequality for divergence is just the interval for convergence that the test gives with the inequality switched and generally isn’t needed. We will usually skip that part.

Now let’s get the interval of convergence. First from the inequality we get,

(x = - sqrt 3 ):
Here the power series is,

This series is divergent by the Divergence Test since (mathop limits_ ight)^n>) doesn’t exist.

(x = sqrt 3 ):
Because we’re squaring the (x) this series will be the same as the previous step.

## Lecture summaries and notes

• 1. Functional notation. Limits. Definition of a derivative. Computing derivatives from the definition. Differentiation rules (nth power, constant multiple, sum, product, chain, quotient). Higher derivatives. Graphing (increasing, decreasing, local extrema, inflection points). 2nd derivative test. Tangent lines/linear approximations. Implicit differentiation. Max-min problems. Related rates problems.

Simmons 1.1-6, 2.1-4, 3.1-3, 3.5-6, 4.1-5, 5.1-2.

Simmons 1.7, 2.5, 3.4, 5.3, 8.1-4, 9.1-5, 10.1-5 (but you can ignore the bits about definite integration for now).

Simmons 6.1-7, 7.1-2, 7.5, 10.1-5 (just for the bits about definite integration, especially 10.2), 10.6.

Lecture notes 3 (.pdf) (I'm still trying to figure out how to make the scan quality better -- if I succeed, I may update these files & let you know in class.)

Lecture notes 5 (.pdf) (Are these scans more readable? Let me know in class if it's an improvement.)

Simmons doesn't have any content on this topic. You might try looking at OCW to see if they have relevant materials in 18.02 if you're interested. (The linear algebra classes 18.06 should cover what we discussed in much greater depth.)

Simmons 17.2, 19.1. Completely optional but possibly of interest: 17.6-7. (17.7 explains how Newton's theory of universal gravitation can be used to prove Kepler's laws of planetary motion.)

The relation $, sin(A)^2+cos(A)^2=I$ also hold for matrices

So we would have to find a matrix for cosine such that $C^2=egin0&-4040&0end$ which is not possible.

is not diagonalizable. If it was $1$ would be the only eigenvalue and $B$ would be equal to the identity matrix.

Therefore if $sin A = B$ , $A$ isn't diagonalizable either. The (complex) Jordan normal form of $A$ would be

egin sin overline A=eginsin lambda & cos lambda & sin lambdaend end

Hence $lambda$ belongs to $pi/2 + 2pi mathbb Z$ and

egin sin overline A=eginsin lambda & 0 & sin lambdaend = egin1 & 0 & 1end end

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## 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.

## 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: 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
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

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?

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