4.13.E: More Problems on Series of Functions

Exercise $\PageIndex{1}$

Verify Note 3 and Example $(\mathrm{c})$ in detail.

Exercise $\PageIndex{2}$

Show that the so-called hyperharmonic series of order $p$,
$$\sum \frac{1}{n^{p}} \quad\left(p \in E^{1}\right) ,$$
converges iff $p>1$.
[Hint: If $p \leq 1$,
$$\sum_{n=1}^{\infty} \frac{1}{n^{p}} \geq \sum_{n=1}^{\infty} \frac{1}{n}=+\infty \quad(\text { Example }(\mathrm{b})) .$$
If $p>1$,
$$\begin{aligned} \sum_{n=1}^{\infty} \frac{1}{n^{p}} &=1+\left(\frac{1}{2^{p}}+\frac{1}{3^{p}}\right)+\left(\frac{1}{4^{p}}+\cdots+\frac{1}{7^{p}}\right)+\left(\frac{1}{8^{p}}+\cdots+\frac{1}{15^{p}}\right)+\cdots \\ & \leq 1+\left(\frac{1}{2^{p}}+\frac{1}{2^{p}}\right)+\left(\frac{1}{4^{p}}+\cdots+\frac{1}{4^{p}}\right)+\left(\frac{1}{8^{p}}+\cdots+\frac{1}{8^{p}}\right)+\cdots \\ &=\sum_{n=0}^{\infty} \frac{1}{\left(2^{p-1}\right)^{n}} . \end{aligned}$$
a convergent geometric series. Explain each step.]

Exercise $\PageIndex{3}$

$\Rightarrow 3.$ Prove the refined comparison test:
(i) If two series of constants, $\sum\left|a_{n}\right|$ and $\sum\left|b_{n}\right|,$ are such that the sequence $\left\{\left|a_{n}\right| /\left|b_{n}\right|\right\}$ is bounded in $E^{1},$ then
$$\sum_{n=1}^{\infty}\left|b_{n}\right|<+\infty \text { implies } \sum_{n=1}^{\infty}\left|a_{n}\right|<+\infty .$$
(ii) If
$$0<\lim _{n \rightarrow \infty} \frac{\left|a_{n}\right|}{\left|b_{n}\right|}<+\infty ,$$
then $\sum\left|a_{n}\right|$ converges if and only if $\sum\left|b_{n}\right|$ does.
What is
$$\lim _{n \rightarrow \infty} \frac{\left|a_{n}\right|}{\left|b_{n}\right|}=+\infty ?$$
[Hint: If (\forall n)|a_{n}| / |b_{n}| \leq K\), then $|a_{n}| \leq K |b_{n}|.$]

Exercise $\PageIndex{4}$

Test $\sum a_{n}$ for absolute convergence in each of the following. Use Problem 3 or Theorem 2 or the indicated references.
(i) $a_{n}=\frac{n+1}{\sqrt{n^{4}+1}}\left(\text { take } b_{n}=\frac{1}{n}\right)$;
(ii) $a_{n}=\frac{\cos n}{\sqrt{n^{3}-1}}\left(\text { take } b_{n}=\frac{1}{\sqrt{n^{3}}} ; \text { use Problem } 2\right)$;
(iii) $a_{n}=\frac{(-1)^{n}}{n^{p}}(\sqrt{n+1}-\sqrt{n}), p \in E^{1}$;
(iv) $a_{n}=n^{5} e^{-n}$ (use Problem 18 of Chapter 3, §15);
(v) $a_{n}=\frac{2^{n}+n}{3^{n}+1}$;
(vi) $a_{n}=\frac{(-1)^{n}}{(\log n)^{q}} ; n \geq 2$;
(vii) $a_{n}=\frac{(\log n)^{q}}{n\left(n^{2}+1\right)}, q \in E^{1}$.
[Hint for (vi) and (vii): From Problem 14 in §2, show that
$$\lim _{y \rightarrow+\infty} \frac{y}{(\log y)^{q}}=+\infty$$
and hence
$$\lim _{n \rightarrow \infty} \frac{(\log n)^{q}}{n}=0 .$$
Then select $b_{n}.$

Exercise $\PageIndex{5}$

Prove that $\sum_{n=1}^{\infty} \frac{n^{n}}{n !}=+\infty$.
[Hint: Show that $n^{n} / n !$ does not tend to 0.]

Exercise $\PageIndex{6}$

Prove that $\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0$.
[Hint: Use Example (d) and Theorem 4.]

Exercise $\PageIndex{7}$

Use Theorems $3,5,6,$ and 7 to show that $\sum\left|f_{n}\right|$ converges uniformly on $B,$ provided $f_{n}(x)$ and $B$ are as indicated below, with $0<a<+\infty$ and $b \in E^{1}.$ For parts $(\text { ix })-(\text { xii }),$ find $M_{n}=\max _{x \in B}\left|f_{n}(x)\right|$ and use Theorem $3 .$ (Calculus rules for maxima are assumed known.)
(i) $\frac{x^{2 n}}{(2 n) !} ;[-a, b]$.
(ii) $(-1)^{n+1} \frac{x^{2 n-1}}{(2 n-1) !} ;[-a, b]$.
(iii) $\frac{x^{n}}{n^{n}} ;[-a, a]$.
(iv) $n^{3} x^{n} ;[-a, a](a<1)$.
$\left.\text { (v) } \frac{\sin n x}{n^{2}} ; B=E^{1} \text { (use Problem } 2\right)$.
(vi) $e^{-n x} \sin n x ;[a,+\infty)$.
(vii) $\frac{\cos n x}{\sqrt{n^{3}+1}} ; B=E^{1}$.
(viii) $a_{n} \cos n x,$ with $\sum_{n=1}^{\infty}\left|a_{n}\right|<+\infty ; B=E^{1}$.
(ix) $x^{n} e^{-n x} ;[0,+\infty)$.
(x) $x^{n} e^{n x} ;\left(-\infty, \frac{1}{2}\right]$.
(xi) $(x \cdot \log x)^{n}, f_{n}(0)=0 ;\left[-\frac{3}{2}, \frac{3}{2}\right]$.
(xii) $\left(\frac{\log x}{x}\right)^{n} ;[1,+\infty)$.
(xiii) $\frac{q(q-1) \cdots(q-n+1) x^{n}}{n !}, q \in E^{1} ;\left[-\frac{1}{2}, \frac{1}{2}\right]$.

Exercise $\PageIndex{8}$

$\Rightarrow 8.$ (Summation by parts.) Let $f_{n}, h_{n},$ and $g_{n}$ be real or complex functions (or let $f_{n}$ and $h_{n}$ be scalar valued and $g_{n}$ be vector valued). Let $f_{n}=$ $h_{n}-h_{n-1}(n \geq 2) .$ Verify that $(\forall m>n>1)$
$$\begin{aligned} \sum_{k=n+1}^{m} f_{k} g_{k} &=\sum_{k=n+1}^{m}\left(h_{k}-h_{k-1}\right) g_{k} \\ &=h_{m} g_{m}-h_{n} g_{n+1}-\sum_{k=n+1}^{m-1} h_{k}\left(g_{k+1}-g_{k}\right) . \end{aligned}$$
[Hint: Rearrange the sum.]

Exercise $\PageIndex{9}$

$\Rightarrow 9.$ (Abel's test.) Let the $f_{n}, g_{n},$ and $h_{n}$ be as in Problem $8,$ with $h_{n}=\sum_{i=1}^{n} f_{i} .$ Suppose that
(i) the range space of the $g_{n}$ is complete;
(ii) $\left|g_{n}\right| \rightarrow 0$ (uniformly) on a set $B ;$ and
(iii) the partial sums $h_{n}=\sum_{i=1}^{n} f_{i}$ are uniformly bounded on $B ;$ i.e.,
$$\left(\exists K \in E^{1}\right)(\forall n) \quad\left|h_{n}\right|<K \text { on } B .$$
Then prove that $\sum f_{k} g_{k}$ converges uniformly on $B$ if $\sum\left|g_{n+1}-g_{n}\right|$ does.
$\text { (This always holds if the } g_{n} \text { are real and } g_{n} \geq g_{n+1} \text { on } B .)$
[Hint: Let $\varepsilon>0 .$ Show that
$$(\exists k)(\forall m>n>k) \quad \sum_{i=n+1}^{m}\left|g_{i+1}-g_{i}\right|<\varepsilon \text { and }\left|g_{n}\right|<\varepsilon \text { on } B .$$
Then use Problem 8 to show that
$$\left|\sum_{i=n+1}^{m} f_{i} g_{i}\right|<3 K \varepsilon .$$
Apply Theorem $3^{\prime}$ of §12.]

Exercise $\PageIndex{9'}$

$\Rightarrow \mathbf{9}^{\prime}.$ Prove that if $\sum a_{n}$ is a convergent series of constants $a_{n} \in E^{1}$ and if $\left\{b_{n}\right\}$ is a bounded monotone sequence in $E^{1},$ then $\sum a_{n} b_{n}$ converges.
[Hint: Let $b_{n} \rightarrow b$. Write
$$a_{n} b_{n}=a_{n}\left(b_{n}-b\right)+a_{n} b$$
$\left.\text { and use Problem } 9 \text { with } f_{n}=a_{n} \text { and } g_{n}=b_{n}-b .\right]$

Exercise $\PageIndex{10}$

$\Rightarrow 10.$ Prove the Leibniz test for alternating series: If $\left\{b_{n}\right\} \downarrow$ and $b_{n} \rightarrow 0$ in $E^{1},$ then $\sum(-1)^{n} b_{n}$ converges, and the sum $\sum_{n=1}^{\infty}(-1)^{n} b_{n}$ differs from $s_{n}=\sum_{k=1}^{n}(-1)^{k} b_{k}$ by $b_{n+1}$ at most.

Exercise $\PageIndex{11}$

$\Rightarrow 11.$ (Dirichlet test.) Let the $f_{n}, g_{n},$ and $h_{n}$ be as in Problem 8 with $\sum_{n=0}^{\infty} f_{n}$ uniformly convergent on $B$ to a function $f,$ and with
$$h_{n}=-\sum_{i=n+1}^{\infty} f_{i} \text { on } B .$$
Suppose that
(i) the range space of the $g_{n}$ is complete; and
(ii) there is $K \in E^{1}$ such that
$$\left|g_{0}\right|+\sum_{n=0}^{\infty}\left|g_{n+1}-g_{n}\right|<K \text { on } B .$$
Show that $\sum f_{n} g_{n}$ converges uniformly on $B$.
[Proof outline: We have
$$\left|g_{n}\right|=\left|g_{0}+\sum_{i=0}^{n-1}\left(g_{i+1}-g_{i}\right)\right| \leq\left|g_{0}\right|+\sum_{i=0}^{n-1}\left|g_{i+1}-g_{i}\right|<K \quad \text { by (ii). }$$
Also,
$$\left|h_{n}\right|=\left|\sum_{i=0}^{n} f_{i}-f\right| \rightarrow 0 \text { (uniformly) on } B$$
by assumption. Hence
$$(\forall \varepsilon>0)(\exists k)(\forall n>k) \quad\left|h_{n}\right|<\varepsilon \text { on } B .$$
Using Problem $8,$ obtain
$$(\forall m>n>k)\left|\sum_{i=n+i}^{m} f_{i} g_{i}\right|<2 K \varepsilon .]$$

Exercise $\PageIndex{12}$

Prove that if $0<p \leq 1$, then $\sum \frac{(-1)^{n}}{n^{p}}$ converges conditionally.
[Hint: Use Problems 11 and 2 .]

Exercise $\PageIndex{13}$

$\Rightarrow 13.$ Continuing Problem 14 in §12, prove that if $\sum\left|f_{n}\right|$ and $\sum\left|g_{n}\right|$ converge on $B$ (pointwise or uniformly), then so do the series
$$\sum\left|a f_{n}+b g_{n}\right|, \sum\left|f_{n} \pm g_{n}\right|, \text { and } \sum\left|a f_{n}\right| .$$
$\left[\text { Hint } :\left|a f_{n}+b g_{n}\right| \leq|a|\left|f_{n}\right|+|b|\left|g_{n}\right| . \text { Use Theorem } 2 .\right]$
For the rest of the section, we define
$$x^{+}=\max (x, 0) \text { and } x^{-}=\max (-x, 0) .$$

Exercise $\PageIndex{14}$

$\Rightarrow 14.$ Given $\left\{a_{n}\right\} \subset E^{*}$ show the following:
(i) $\sum a_{n}^{+}+\sum a_{n}^{-}=\sum\left|a_{n}\right|$.
(ii) If $\sum a_{n}^{+}<+\infty$ or $\sum a_{n}^{-}<+\infty,$ then $\sum a_{n}=\sum a_{n}^{+}-\sum a_{n}^{-}$.
(iii) If $\sum a_{n}$ converges conditionally, then $\sum a_{n}^{+}=+\infty=\sum a_{n}^{-}$.
(iv) If $\sum\left|a_{n}\right|<+\infty,$ then for any $\left\{b_{n}\right\} \subset E^{1}$,
$$\sum\left|a_{n} \pm b_{n}\right|<+\infty \text { iff } \sum\left|b_{n}\right|<\infty ;$$
$$\text { moreover, } \sum a_{n} \pm \sum b_{n}=\sum\left(a_{n} \pm b_{n}\right) \text { if } \sum b_{n} \text { exists. }$$
$\left.\text { [Hint: Verify that }\left|a_{n}\right|=a_{n}^{+}+a_{n}^{-} \text {and } a_{n}=a_{n}^{+}-a_{n}^{-} . \text {Use the rules of } §4 .\right]$

Exercise $\PageIndex{15}$

$\Rightarrow 15.$ (Abel's theorem.) Show that if a power series
$$\sum_{n=0}^{\infty} a_{n}(x-p)^{n} \quad\left(a_{n} \in E, x, p \in E^{1}\right)$$
converges for some $x=x_{0} \neq p,$ it converges uniformly on $\left[p, x_{0}\right]$ (or $\left.\left[x_{0}, p\right] \text { if } x_{0}<p\right)$.
[Proof outline: First let $p=0$ and $x_{0}=1 .$ Use Problem 11 with
$$f_{n}=a_{n} \text { and } g_{n}(x)=x^{n}=(x-p)^{n} .$$
As $f_{n}=a_{n} 1^{n}=a_{n}\left(x_{0}-p\right)^{n}$, the series $\sum f_{n}$ converges by assumption. The convergence is uniform since the $f_{n}$ are constant. Verify that if $x=1$, then
$$\sum_{k=1}^{\infty}\left|g_{k+1}-g_{k}\right|=0 ,$$
and if $0 \leq x<1,$ then
$$\sum_{k=0}^{\infty}\left|g_{k+1}-g_{k}\right|=\sum_{k=0}^{\infty} x^{k}|x-1|=(1-x) \sum_{k=0}^{\infty} x^{k}=1 \quad \text { (a geometric series). }$$
Also, $\left|g_{0}(x)\right|=x^{0}=1 .$ Thus by Problem $11(\text { with } K=2), \sum f_{n} g_{n}$ converges uniformly on $[0,1],$ proving the theorem for $p=0$ and $x_{0}=1 .$ The general case reduces to this case by the substitution $x-p=\left(x_{0}-p\right) y .$ Verify!]

Exercise $\PageIndex{16}$

Prove that if
$$0<\underline{\lim} a_{n} \leq \overline{\lim } a_{n}<+\infty ,$$
then the convergence radius of $\sum a_{n}(x-p)^{n}$ is 1.

Exercise $\PageIndex{17}$

Show that a conditionally convergent series $\sum a_{n}\left(a_{n} \in E^{1}\right)$ can be rearranged so as to diverge, or to converge to any prescribed sum $s$. [Proof for $s \in E^{1} :$ Using Problem 14(iii), take the first partial sum
$$a_{1}^{+}+\cdots+a_{m}^{+}>s .$$
Then adjoin terms
$$-a_{1}^{-},-a_{2}^{-}, \ldots,-a_{n}^{-}$$
until the partial sum becomes less than $s .$ Then add terms $a_{k}^{+}$ until it exceeds $s$. Then adjoin terms $-a_{k}^{-}$ until it becomes less than $s,$ and so on.
As $a_{k}^{+} \rightarrow 0$ and $a_{k}^{-} \rightarrow 0$ (why?), the rearranged series tends to $s .$ (Why?)
Give a similar proof for $s=\pm \infty$. Also, make the series oscillate, with no sum.]

Exercise $\PageIndex{18}$

Prove that if a power series $\sum a_{n}(x-p)^{n}$ converges at some $x=x_{0} \neq p$, it converges absolutely (pointwise) on $G_{p}(\delta)$ if $\delta \leq\left|x_{0}-p\right|$.
[Hint: By Theorem $6, \delta \leq\left|x_{0}-p\right| \leq r\left(r=\text { convergence radius). Fix any } x \in G_{p}(\delta) .\right.$ Show that the line $\overrightarrow{p x},$ when extended, contains a point $x_{1}$ such that $|x-p|< \left|x_{1}-p\right|<\delta \leq r .$ By Theorem $6,$ the series converges absolutely at $x_{1},$ hence at $x$ as well, by Theorem 7.]