2.7 Infinite series

Probably the most interesting and important examples of sequences are those that arise as the partial sums of an infinite series.In fact, it will be infinite series that allow us to explain such things as trigonometric and exponential functions.

Let ${\left\{a_n\right\}}_0^{\infty }$ be a sequence of real or complex numbers. By the infinite series $\sum a_n$ we mean the sequence $\left\{S_N\right\}$ defined by

$$S_N=\sum _{n=0}^N{a}_n.$$

The sequence $\left\{S_N\right\}$ is called the sequence of partial sums of the infinite series $\sum a_n,$ and the infinite series is saidto be summable to a number $S,$ or to be convergent, if the sequence $\left\{S_N\right\}$ of partial sums converges to $S.$ The sum of an infinite series is the limit of its partial sums.

An infinite series $\sum a_n$ is called absolutely summable or absolutely convergent if the infinite series $\sum |a_n|$ is convergent.

If $\sum a_n$ is not convergent, it is called divergent . If it is convergent but not absolutely convergent, it iscalled conditionally convergent .

A few simple formulas relating the $a_n$ 's and the $S_N$ 's are useful:

$$S_N=a_0+a_1+a_2+...+a_N,$$

$$S_{N+1}=S_N+a_{N+1},$$

and

$$S_M-S_K=\sum _{n=K+1}^M{a}_n=a_{K+1}+a_{K+2}+...+A_M,$$

for $M>K.$

REMARK Determining whether or not a given infinite series converges is one of the most important and subtle parts of analysis.Even the first few elementary theorems depend in deep ways on our previous development, particularly the Cauchy criterion.

Let $\left\{a_n\right\}$ be a sequence of nonnegative real numbers. Then the infinite series $\sum a_n$ is summable if and only if the sequence $\left\{S_N\right\}$ of partial sums is bounded.

If $\sum a_n$ is summable, then $\left\{S_N\right\}$ is convergent, whence bounded according to [link] . Conversely, we see from the hypothesis that each $a_n\ge 0$ that $\left\{S_N\right\}$ is nondecreasing ( $S_{N+1}=S_N+a_{N+1}\ge S_N$ ). So, if $\left\{S_N\right\}$ is bounded, then it automatically converges by [link] , and hence the infinite series $\sum a_n$ is summable.

The next theorem is the first one most calculus students learn about infinite series. Unfortunately, it is often misinterpreted, so be careful!Both of the proofs to the next two theorems use [link] , which again is a serious and fundamental result about the real numbers.Therefore, these two theorems must be deep results themselves.

Let $\sum a_n$ be a convergent infinite series. Then the sequence $\left\{a_n\right\}$ is convergent, and $lim{a}_n=0.$

Because $\sum a_n$ is summable, the sequence $\left\{S_N\right\}$ is convergent and so is a Cauchy sequence. Therefore, given an $ϵ>0,$ there exists an $N_0$ so that $|S_n-S_m|<ϵ$ whenever both $n$ and $m\ge N_0.$ If $n>N_0,$ let $m=n-1.$ We have then that $|a_n|=|S_n-S_m|<ϵ,$ which completes the proof.

REMARK Note that this theorem is not an “if and only if” theorem. The harmonic series (part (b) of [link] below) is the standard counterexample.The theorem above is mainly used to show that an infinite series is not summable. If we can prove that the sequence $\left\{a_n\right\}$ does not converge to 0, then the infinite series $\sum a_n$ does not converge. The misinterpretation of this result referred to above is exactly intrying to apply the (false) converse of this theorem.