2.7 Infinite series (Page 2/3)

Analysis of functions of a single Page 2 / 3

If $\sum a_{n}$ is an absolutely convergent infinite series of complex numbers, then it is a convergent infinite series.(Absolute convergence implies convergence.)

If ${S_{N}}$ denotes the sequence of partial sums for $\sum a_{n},$ and if ${T_{N}}$ denotes the sequence of partial sums for $\sum | a_{n} |,$ then

| S_{M} - S_{N} | = | \sum_{n = N + 1}^{M} a_{n} | \leq \sum_{n = N + 1}^{M} | a_{n} | = | T_{M} - T_{N} |

for all $N$ and $M .$ We are given that ${T_{N}}$ is convergent and hence it is a Cauchy sequence. So, by the inequality above, ${S_{N}}$ must also be a Cauchy sequence. (If $| T_{N} - T_{M} | < ϵ,$ then $| S_{N} - S_{M} | < ϵ$ as well.) This implies that $\sum a_{n}$ is convergent.

Let $z$ be a complex number, and define a sequence ${a_{n}}$ by $a_{n} = z^{n} .$ Consider the infinite series $\sum a_{n} .$ Show that $\sum_{n = 0}^{\infty} a_{n}$ converges to a number $S$ if and only if $| z | < 1 .$ Show in fact that $S = 1 / (1 - z),$ when $| z | < 1 .$

HINT: Evaluate explicitly the partial sums $S_{N},$ and then take their limit. Show that $S_{N} = \frac{1 - z^{N + 1}}{1 - z} .$

Show that $\sum_{n = 1}^{\infty} \frac{1}{n (n + 1)}$ converges to $1,$ by computing explicit formulas for the partial sums. HINT: Use a partial fraction decomposition for the $a_{n}$ 's.
(The Harmonic Series.) Show that $\sum_{n = 1}^{\infty} 1 / n$ diverges by verifying that $S_{2^{k}} > k / 2 .$ HINT: Group the terms in the sum as follows,
$1 + \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) + (\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}) + (\frac{1}{9} + \frac{1}{10} + ... + \frac{1}{16}) + ...,$
and then estimate the sum of each group. Remember this example as an infinite series that diverges, despite thefact that is terms tend to 0.

The next theorem is the most important one we have concerning infinite series of numbers.

Comparison test

Suppose ${a_{n}}$ and ${b_{n}}$ are two sequences of nonnegative real numbers for which there exists a positive integer $M$ and a constant $C$ such that $b_{n} \leq C a_{n}$ for all $n \geq M .$ If the infinite series $\sum a_{n}$ converges, so must the infinite series $\sum b_{n} .$

We will show that the sequence ${T_{N}}$ of partial sums of the infinite series $\sum b_{n}$ is a bounded sequence. Then, by [link] , the infinite series $\sum b_{n}$ must be summable.

Write $S_{N}$ for the $N$ th partial sum of the convergent infinite series $\sum a_{n} .$ Because this series is summable, its sequence of partial sums is a bounded sequence. Let B be a number such that $S_{N} \leq B$ for all $N .$ We have for all $N > M$ that

\begin{matrix} r c l T_{N} & = & \sum_{n = 1}^{N} b_{n} \\ = & \sum_{n = 1}^{M} b_{n} + \sum_{n = M + 1}^{N} b_{n} \\ \leq & \sum_{n = 1}^{M} b_{n} + \sum_{n = M + 1}^{N} C a_{n} \\ = & \sum_{n = 1}^{M} b_{n} + C \sum_{n = M + 1}^{N} a_{n} \\ \leq & \sum_{n = 1}^{M} b_{n} + C \sum_{n = 1}^{N} a_{n} \\ \leq & \sum_{n = 1}^{M} b_{n} + C S_{N} \\ \leq & \sum_{n = 1}^{M} b_{n} + C B, \end{matrix}

which completes the proof, since this final quantity is a fixed constant.

Let ${a_{n}}$ and ${b_{n}}$ be as in the preceding theorem. Show that if $\sum b_{n}$ diverges, then $\sum a_{n}$ also must diverge.
Show by example that the hypothesis that the $a_{n}$ 's and $b_{n}$ 's of the Comparison Test are nonnegative can not be dropped.

Let ${a_{n}}$ be a sequence of positive numbers.

If $lim sup a_{n + 1} / a_{n} < 1,$ show that $\sum a_{n}$ converges. HINT: If $lim sup a_{n + 1} / a_{n} = α < 1,$ let $β$ be a number for which $α < β < 1 .$ Using part (a) of [link] , show that there exists an $N$ such that for all $n > N$ we must have $a_{n + 1} / a_{n} < β,$ or equivalently $a_{n + 1} < β a_{n},$ and therefore $a_{N + k} < β^{k} a_{N} .$ Now use the comparison test with the geometric series $\sum β^{k} .$
If $lim inf a_{n + 1} / a_{n} > 1,$ show that $\sum a_{n}$ diverges.
As special cases of parts (a) and (b), show that ${a_{n}}$ converges if ${lim}_{n} a_{n + 1} / a_{n} < 1,$ and diverges if ${lim}_{n} a_{n + 1} / a_{n} > 1 .$
Find two examples of infinite series' $\sum a_{n}$ of positive numbers, such that $lim a_{n + 1} / a_{n} = 1$ for both examples, and such that one infinite series converges and the other diverges.

Derive the Root Test: If ${a_{n}}$ is a sequence of positive numbers for which $lim sup a_{n}^{1 / n} < 1,$ then $\sum a_{n}$ converges. And, if $lim inf a_{n}^{1 / n} > 1,$ then $\sum a_{n}$ diverges.
Let $r$ be a positive integer. Show that $\sum 1 / n^{r}$ converges if and only if $r \geq 2 .$ HINT: Use [link] and the Comparison Test for $r = 2 .$
Show that the following infinite series are summable.
$\sum 1 / (n^{2} + 1), \sum n / 2^{n}, \sum a^{n} / n!,$
for $a$ any complex number.

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Source: OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1

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