Is there any comprehensive list (books, online, ...) of rules for manipulating infinite series (partial sums) to find convergence of a sum? Often authors use some "trick" to compute an infinite series. Following this trick is always a disclaimer, such as "adding infinite sequences is not the same as adding discrete values so the usual rules of algebra will not work."

Huh? Then how am I supposed to learn what I can and cannot do when all I have been shown is a trick that works in a particular case? For example,

\begin{align*} \sum_{n=1}^{\infty}\frac{1}{n(n+1)}\rightarrow S_{n}&=\sum_{k=1}^{n}\frac{1}{k(k+1)}\\\ &=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\frac{1}{n(n+1)}\\\ &=\; \sum_{k=1}^{n}(\frac{1}{k}-\frac{1}{k+1})\\ &=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\cdots+(\frac{1}{n}-\frac{1}{n+1})\\\ &=1-\frac{1}{n+1}\rightarrow \lim_{n\rightarrow \infty}\left(1-\frac{1}{n+1}\right)=1-0=1. \end{align*} OK, I see the author used partial fractions expansion followed by grouping like-terms. However, elsewhere I'll find that grouping like-terms will lead you to the wrong answer. Such is the case with this infinite sum:

$$ S_{n}=1-1+1-1+1+\cdots \stackrel{?}{\Longrightarrow} (1-1)+(1-1)+\cdots=0 $$ $$ S_{n}=1-1+1-1+1+\cdots\stackrel{?}{\Longrightarrow} 1+(-1+1)+(-1+1)+\cdots=1 $$ ...So does this mean infinite series are not associative? Or are they? What are the invariant properties of infinite series that can be confidently used when manipulating an infinite series?

FOLLOW UP: Thanks for the great answers so far, yet they have led me to question the practicality of writing in summation form. Is it not more useful to simply write the infinite sum as its corresponding infinite partial sum sequence?

E.g.: Why is the fourier series written as an infinite sum? I realize it can be used to create a continuous analog for many types of discontinuous functions, and yet I feel I gain nothing when I write out the fourier series to say, a solution to a partial differential equation, because Idk how to evaluate the output to a particular input since its defined by an infinite sum of sin's and cosines. (Besides finding the partial sum sequence) Is there not a more direct way?

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    $\begingroup$ An infinite sum is always taken as a limit of the sequence of partial sums. If you want to evaluate $\sum_{k=1}^{\infty} a_k$, then compute $S_n = \sum_{k=1}^{n} a_k$ and look at $\lim_{n \rightarrow \infty} S_n$. For the first series you have, $\sum_{k=1}^{\infty} \frac1{k(k+1)}$. Consider $S_n = \sum_{k=1}^{n} \frac1{k(k+1)} = 1 - \frac1{n+1}$. $\lim_{n \rightarrow \infty} S_n = 1$ and hence we say the series converges to $1$. $\endgroup$ – user17762 Dec 20 '11 at 17:46
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    $\begingroup$ In the second example, $\sum_{k=0}^{\infty} (-1)^k$, consider the partial sum $S_n = \sum_{k=0}^{n} (-1)^k$. We then get $S_n = \frac{1+(-1)^n}{2}$ for which the limit doesn't exist since $(-1)^n$ oscillates between $+1$ and $-1$. Hence, you can always do such "grouping" (i.e. make use of associativity, commutativity) while dealing with finite sums to simplify $S_n$. But once we have $S_n$ to evaluate the series you need to look at the limit of $S_n$. $\endgroup$ – user17762 Dec 20 '11 at 17:51
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    $\begingroup$ Summing infinite times is just impossible, so when one encounters an infinite summation symbol $$ \sum_{n=1}^{\infty} a_n, $$ we have to give a precise meaning to it, which we call a summability method. The most general summability method is, of course, to consider it as the limit of its partial sums. But in some circumstances we give other types of summability methods, such as Cesaro summability and Abel summability. For example, there are occasions where we have to compute divergent series like $1^3 + 2^3 + 3^3 + \cdots$, which certainly requires an exotic kind of regularization method. $\endgroup$ – Sangchul Lee Dec 20 '11 at 18:04
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    $\begingroup$ @skyfire: Any infinite sum including the Fourier series is taken as the limit of the partial sums. If you have $f(x) = \sum_{n=-\infty}^{\infty} a_n e^{inx}$, what it means is if $f_N(x) = \sum_{n=-N}^{N} a_n e^{inx}$, then $||f(x) - f_N(x)||_2 \rightarrow 0$ as $N \rightarrow \infty$. (As an aside, not "all" functions can be written as a Fourier series) $\endgroup$ – user17762 Dec 20 '11 at 18:46

Infinite series bugged mathematicians and thinkers for a long time. You can read a little about Zeno's Paradoxes, particularily, the dichotomy paradox.

I guess important questions were

  1. Why would an infinite amount of positive quantities be finite?
  2. How can we assingn a sum to an infinite amount of numbers, and to a divergent series?
  3. Do the known arithmetics work for infinite series?
  4. If the sum is finite, how can we find the exact value of the sum?
  5. Does the order of the sum matter?

Many of this questions were answered, and some criteria was established such as

  1. What an infinite series is.
  2. Converge and divergence of a series. D'Alabert, Cauchy, Raabe, Kummer and Gauss provided several methods to sort this out.
  3. The concept of absolute convergence and conditional convergence
  4. The uniform convergence of a series of functions.
  5. The sum of alternating series.

You will find particularily helpful to read about conditional and aboslute convergence: basically, if a series is absolutely convergent, we can manipulate it with the ordinary arithmetic, but if it is conditionally convergent, we can make it sum to any number we want, or make it diverge to infinity. For a divergent series, the normal arithmetic fails, basically, because we're thinking about infinity as a number, when it isn't one. For example, let $S$ denote the infinite sum:

$$S = 1+2+4+8+\cdots$$

We can note that

$$S = 1+2(1+2+4+\cdots)$$

that is

$$S = 1+2S$$

which means

$$-S = 1 $$ or

$$S=-1 \text{ (!)}$$

In the usual sense of arithmetic and algebra, if we are summing infinite many positive numbers, we'd at least expect the sum to be positive, but if we apply the usual rules of arithmetic, we find it's actually $-1$. What does this tell us?

  1. $\infty$ is not a number.
  2. If we want to make sums like $S$ meaningful, we need to develop a new theory for infinite divergent sums.
  • $\begingroup$ just what I was looking for $\endgroup$ – skyfire Mar 2 '12 at 22:25
  • $\begingroup$ @skyfire Glad to help. $\endgroup$ – Pedro Tamaroff Mar 2 '12 at 22:26

I would suggest you read the book "An introduction to the Theory of Infinite Series" by Thomas Bromwich.

P.S.: There is a tale that goes something like Ramanujan was advised by British Mathematicians to read this book when he wanted to prove that the sum of a series of positive terms is negative ( I neither remember the series, nor Ramanujan's "sum" but all I could remember is it was divergent").

An online version, downloadable in many different formats, is here on the archive.com .

Read Comment below by Sivaram where he points out both the series and the sum!

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    $\begingroup$ The series was $1+2+3+\cdots$ which ramanujam regularized to get $\zeta(-1) = -1/12$ $\endgroup$ – user17762 Dec 20 '11 at 18:05

The classical reference is Konrad Knopp, Infinite Sequences and Series :-) Maybe outdated today.


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