2.7 Infinite series  (Page 2/3)

Let $z$ be a complex number, and define a sequence $\left\{a_n\right\}$ 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 $\left|z\right|<1.$ Show in fact that $S=1/(1-z),$ when $\left|z\right|<1.$

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

1. 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.
2. (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.

1. Let $\left\{a_n\right\}$ and $\left\{b_n\right\}$ be as in the preceding theorem. Show that if $\sum b_n$ diverges, then $\sum a_n$ also must diverge.
2. 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 $\left\{a_n\right\}$ be a sequence of positive numbers.

1. If $lim\; sup{a}_{n+1}/a_n<1,$ show that $\sum a_n$ converges. HINT: If $lim\; sup{a}_{n+1}/a_n=\alpha <1,$ let $\beta$ be a number for which $\alpha <\beta <1.$ Using part (a) of [link] , show that there exists an $N$ such that for all $n>N$ we must have $a_{n+1}/a_n<\beta ,$ or equivalently $a_{n+1}<\beta a_n,$ and therefore $a_{N+k}<{\beta }^k{a}_N.$ Now use the comparison test with the geometric series $\sum {\beta }^k.$
2. If $lim\; inf{a}_{n+1}/a_n>1,$ show that $\sum a_n$ diverges.
3. As special cases of parts (a) and (b), show that $\left\{a_n\right\}$ converges if ${lim}_n{a}_{n+1}/a_n<1,$ and diverges if ${lim}_n{a}_{n+1}/a_n>1.$
4. 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.

1. Derive the Root Test: If $\left\{a_n\right\}$ 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.
2. Let $r$ be a positive integer. Show that $\sum 1/n^r$ converges if and only if $r\ge 2.$ HINT: Use [link] and the Comparison Test for $r=2.$
3. 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.