Real Analysis

John M. Howie CBE, MA, D.Phil, DSc, Hon D.Univ, FRSE

Chapter 7 Sequences and Series of Functions - all with Video Answers

Educators

Chapter Questions

Problem 1

Suppose that $\left(f_n\right) \rightarrow f$ and $\left(g_n\right) \rightarrow g$ uniformly on $[a, b]$, and that $f, g$ are bounded functions.
a) Show that $\left(f_n+g_n\right) \rightarrow f+g$ uniformly on $[a, b]$.
b) Show that $\left(f_n \cdot g_n\right) \rightarrow f \cdot g$ uniformly on $[a, b]$.
c) Suppose that $f_n(x)$ is non-zero for all $x$ and for all $n$, and that there exists $\delta>0$ with the property that $f(x) \geq \delta$ for all $x$ in $[a, b]$. Show that $\left(1 / f_n\right) \rightarrow 1 / f$ uniformly in $[a, b]$.

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Problem 2

Find a sequence of functions, each discontinuous at every point in $[0,1]$, converging uniformly to a continuous function.

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Problem 3

Let
$$
f_n(x)=n^2 x^n(1-x) \quad(n \in \mathbb{N}, x \in[0,1]) .
$$

Show that $\left(f_n\right) \rightarrow 0$ pointwise in $[0,1]$, but that the convergence is not uniform. Show also that
$$
\lim _{n \rightarrow \infty} \int_0^1 f_n \neq \int_0^1\left(\lim _{n \rightarrow \infty} f_n(x)\right) d x .
$$

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Problem 4

Let
$$
f_n(x)=n x^n(1-x) \quad(n \in \mathbb{N}, x \in[0,1]) .
$$

Show that $\left(f_n\right) \rightarrow 0$ pointwise in $[0,1]$, but that the convergence is not uniform. Show, however, that
$$
\lim _{n \rightarrow \infty} \int_0^1 f_n=\int_0^1\left(\lim _{n \rightarrow \infty} f_n(x)\right) d x
$$

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Problem 5

Let
$$
f_n(x)=x^n(1-x) \quad(n \in \mathbb{N}, x \in[0,1]) .
$$

Show that $\left(f_n\right) \rightarrow 0$ uniformly in $[0,1]$, but that $\left(f_n^{\prime}\right)$ is not uniformly convergent.

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Problem 6

It is easy to extend the definition of uniform convergence from the case of a closed finite interval to more general subsets of $\mathbb{R}$. Let $f_n:[0, \infty) \rightarrow \mathbb{R}$ and $f:[0, \infty) \rightarrow \mathbb{R}$ be given by
$$
f_n(x)=\frac{x}{x+n}, \quad f(x)=0 .
$$
a) Show that $\left(f_n\right) \rightarrow f$ pointwise in $[0, \infty)$.
b) Show that, for each $b>0,\left(f_n\right) \rightarrow f$ uniformly in $[0, b]$.
c) Show that $\left(f_n\right)$ does not converge uniformly to $f$ in $[0, \infty)$.

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Problem 7

Let $f_n:[0, \infty) \rightarrow \mathbb{R}$ and $f:[0, \infty) \rightarrow \mathbb{R}$ be given by
$$
f_n(x)=\frac{n x}{1+n x}, \quad f(x)= \begin{cases}0 & \text { if } x=0 \\ 1 & \text { otherwise }\end{cases}
$$
a) Show that $\left(f_n\right) \rightarrow f$ pointwise in $[0, \infty)$.
b) Show that, for all $b>0,\left(f_n\right) \rightarrow f$ uniformly in $[b, \infty)$.
c) Show that $\left(f_n\right)$ does not converge uniformly to $f$ on $[0, \infty)$.

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Problem 8

Let
$$
f_n(x)=x+\frac{1}{n}, \quad f(x)=x \quad(n \in \mathbb{N}, x \in \mathbb{R}) .
$$

Show that $\left(f_n\right) \rightarrow f$ uniformly in $\mathbb{R}$, but that $f_n^2$ (defined by $\left.\left(f_n^2\right)(x)=(f(x))^2\right)$ does not converge uniformly to $f^2$.

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03:28

Problem 8

Show that
$$
\sum_{n=1}^{\infty} \frac{n x^2}{n^3+x^3}
$$
is uniformly convergent in any finite interval $[0, b]$.

Mengchun Cai

Numerade Educator

Problem 10

Investigate pointwise and uniform convergence for the following series $\sum_{n=1}^{\infty} f_n$. Assume that $x \in[0, \infty)$. If there is uniform convergence only for a subset of $[0, \infty)$, find that subset.
$$
f_n(x)=\frac{x^n}{x^n+1}, \quad f_n(x)=\frac{1}{n^2(x+1)^2}, \quad f_n(x)=\frac{1}{x^n+1} \text {. }
$$

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Problem 11

Investigate the pointwise and uniform convergence of the series
$$
\sum_{n=0}^{\infty} \frac{x^2}{\left(1+x^2\right)^n} \text {. }
$$

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03:28

Problem 12

Determine whether
$$
\sum_{n=1}^{\infty} \frac{x}{n^{3 / 2}+n^{3 / 4} x^2} \text { and } \sum_{n=1}^{\infty} \frac{x}{n^{3 / 4}+n^{3 / 2} x^2}
$$
are uniformly convergent in $[0,1]$.

Mengchun Cai

Numerade Educator

Problem 13

Investigate the pointwise and uniform convergence in $[0,1]$ of the series
$$
\sum_{n=1}^{\infty} \frac{x^n(1-x)}{n^2} \text { and } \sum_{n=1}^{\infty} \frac{x^n(1-x)}{n} \text {. }
$$

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03:28

Problem 14

Show that
$$
\sum_{n=1}^{\infty} \frac{1}{n}(\log (n+x)-\log n)
$$
is uniformly convergent in $[0,1]$.

Mengchun Cai

Numerade Educator

03:27

Problem 15

Find the interval of convergence of $\sum_{n=0}^{\infty} a_n x^n$, where
(i) $a_n=2^n /(n+1)$;
(ii) $a_n=(-1)^n / \sqrt{n+1}$;
(iii) $a_n=n!/(n+1)^n$;
(iv) $a_n=1 /(2+(1 /(n+1)))^n$;
(v) $a_n=1 /(n+2) \log (n+2)$;
(vi) $a_n=(n!)^2 /(2 n)!$.

Nick Johnson

Numerade Educator

Problem 16

Show that, for all $x$ in $(1,1)$,
$$
\sum_{n=1}^{\infty} \frac{n}{n+1} x^{n+1}=\frac{1}{(1-x)}+\log (1-x)-1 .
$$

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01:53

Problem 17

Prove that $\sin 3 x=3 \sin x-4 \sin ^3 x$, and hence find the TaylorMaclaurin series for $\sin ^3 x$.

Subham Jyoti Mishra

Numerade Educator

03:30

Problem 18

Suppose that the power series $\sum_{n=0}^{\infty} a_n x^n$ has radius of convergence $R$. Prove that, for all $|x|<\min \{1, R\}$,
$$
\frac{1}{1-x} \sum_{n=0}^{\infty} a_n x^n=\sum_{n=0}^{\infty} s_n x^n,
$$
where $s_n=a_0+a_1+\cdots+a_n$. Deduce that, for all $|x|<1$,
$$
\frac{1}{1-x} \log (1+x)=x+\left(1-\frac{1}{2}\right) x^2+\left(1-\frac{1}{2}+\frac{1}{3}\right) x^3+\cdots \text {. }
$$

Ahmed Ibrahim

Numerade Educator

02:39

Problem 19

Let $f(x)=\sinh ^{-1} x \quad(x \in \mathbb{R})$. Show that, for all $x$ in $\mathbb{R}$
$$
\left(1+x^2\right) f^{\prime \prime}(x)+x f^{\prime}(x)=0 .
$$

By differentiating $n$ times and equating $x$ to 0 , show that $f^{(n+2)}(0)=$ $-n^2 f^{(n)}(0)$, and deduce that the Taylor-Maclaurin series for $\sinh ^{-1}(x)$ is
$$
\sum_{n=0}^{\infty}(-1)^{n+1} \frac{(2 n)!}{(2 n+1) 2^{2 n}(n!)^2} x^{2 n+1} .
$$

Mirza Aslam Beig

Numerade Educator