Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 4

Series and limits - all with Video Answers

Educators

Chapter Questions

Problem 1

Sum the even numbers between 1000 and 2000 inclusive.

Derrick Danso

Derrick Danso

Numerade Educator

Problem 2

If you invest $£ 1000$ on the first day of each year, and interest is paid at $5 \%$ on your balance at the end of each year, how much money do you have after 25 years?

Eduard Sanchez

Numerade Educator

Problem 3

How does the convergence of the series
$$
\sum_{n=r}^{\infty} \frac{(n-r) !}{n !}
$$
depend on the integer $r$ ?

Derrick Danso

Numerade Educator

Problem 4

Show that for testing the convergence of of the series
$$
x+y+x^{2}+y^{2}+x^{3}+y^{3}+\cdots
$$
where $0<x<y<1$, the D'Alembert ratio test fails but the Cauchy root test is successful.

Eduard Sanchez

Numerade Educator

Problem 5

Find the sum $S_{N}$ of the first $N$ terms of the following series, and hence determine whether the series are convergent, divergent, or oscillatory:
(a) $\sum_{n=1}^{\infty} \ln \left(\frac{n+1}{n}\right)$,
(b) $\sum_{n=0}^{\infty}(-2)^{n}$,
(c) $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{3^{n}}$.

Derrick Danso

Numerade Educator

Problem 6

By grouping and rearranging terms of the absolutely convergent series
$$
S=\sum_{n=1}^{\infty} \frac{1}{n^{2}}
$$
show that
$$
S_{\mathrm{o}}=\sum_{n \text { odd }}^{\infty} \frac{1}{n^{2}}=\frac{3 S}{4}
$$

Eduard Sanchez

Numerade Educator

Problem 7

Use the difference method to sum the series
$$
\sum_{n=2}^{N} \frac{2 n-1}{2 n^{2}(n-1)^{2}}
$$

Derrick Danso

Numerade Educator

Problem 8

The $N+1$ complex numbers $\omega_{m}$ are given by $\omega_{m}=\exp (2 \pi i m / N)$ for $m=$ $0,1,2, \ldots, N$
(a) Evaluate the following:
(i) $\sum_{m=0}^{N} \omega_{m}$,
(ii) $\sum_{m=0}^{N} \omega_{m}^{2}$,
(iii) $\sum_{m=0}^{N} \omega_{m} x^{m}$.
(b) Use these results to evaluate
(i) $\sum_{m=0}^{N}\left[\cos \left(\frac{2 \pi m}{N}\right)-\cos \left(\frac{4 \pi m}{N}\right)\right]$
(ii) $\sum_{m=0}^{3} 2^{m} \sin \left(\frac{2 \pi m}{3}\right)$.

Jarrett Lancaster

Jarrett Lancaster

Numerade Educator

Problem 9

Prove that
$$
\cos \theta+\cos (\theta+\alpha)+\cdots+\cos (\theta+n \alpha)=\frac{\sin \frac{1}{2}(n+1) \alpha}{\sin \frac{1}{2} \alpha} \cos \left(\theta+\frac{1}{2} n \alpha\right)
$$

Derrick Danso

Numerade Educator

Problem 10

Determine whether the following series converge $(\theta$ and $p$ are positive real numbers):
(a) $\sum_{n=1}^{\infty} \frac{2 \sin n \theta}{n(n+1)}$,
(b) $\sum_{n=1}^{\infty} \frac{2}{n^{2}}$,
(c) $\sum_{n=1}^{\infty} \frac{1}{2 n^{1 / 2}}$,
(d) $\sum_{n=2}^{\infty} \frac{(-1)^{n}\left(n^{2}+1\right)^{1 / 2}}{n \ln n}$,
(e) $\sum_{n=1}^{\infty} \frac{n^{p}}{n !}$.

Eduard Sanchez

Numerade Educator

Problem 11

Find the real values of $x$ for which the following are series convergent:
(a) $\sum_{n=1}^{\infty} \frac{x^{n}}{n+1}$,
(b) $\sum_{n=1}^{\infty}(\sin x)^{n}$,
(c) $\sum_{n=1}^{\infty} n^{x}$,
(d) $\sum_{n=1}^{\infty} e^{n x}$,
(e) $\sum_{n=2}^{\infty}(\ln n)^{x}$.

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 12

Determine whether the following series are convergent:
(a) $\sum_{n=1}^{\infty} \frac{n^{1 / 2}}{(n+1)^{1 / 2}}$,
(b) $\sum_{n=1}^{\infty} \frac{n^{2}}{n !}$,
(c) $\sum_{n=1}^{\infty} \frac{(\ln n)^{n}}{n^{n / 2}}$,
(d) $\sum_{n=1}^{\infty} \frac{n^{n}}{n !}$.

Eduard Sanchez

Numerade Educator

Problem 13

Determine whether the following series are absolutely convergent, convergent or oscillatory:
(a) $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{5 / 2}}$
(b) $\sum_{n=1}^{\infty} \frac{(-1)^{n}(2 n+1)}{n}$,
(c) $\sum_{n=0}^{\infty} \frac{(-1)^{n}|x|^{n}}{n !}$,
(d) $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n^{2}+3 n+2}$,
(e) $\sum_{n=1}^{\infty} \frac{(-1)^{n} 2^{n}}{n^{1 / 2}}$.

Derrick Danso

Numerade Educator

Problem 14

Determine the positive values of $x$ for which the following series converges:
$$
\sum_{n=1}^{\infty} \frac{x^{n / 2} e^{-n}}{n}
$$

Eduard Sanchez

Numerade Educator

Problem 15

Prove that
$$
\sum_{n=2}^{\infty} \ln \left[\frac{n^{r}+(-1)^{n}}{n^{r}}\right]
$$
is absolutely convergent for $r=2$, but only conditionally convergent for $r=1$.

Nick Johnson

Numerade Educator

Problem 16

An extension to the proof of the integral test (subsection 4.3.2) shows that, if $f(x)$ is positive, continuous and monotonically decreasing, for $x \geq 1$, and the series $f(1)+f(2)+\cdots$ is convergent, then its sum does not exceed $f(1)+L$, where $L$ is the integral
$$
\int_{1}^{\infty} f(x) d x
$$
Use this result to show that the $\operatorname{sum} \zeta(p)$ of the Riemann zeta series $\sum n^{-p}$, with

Eduard Sanchez

Numerade Educator

Problem 17

Demonstrate that rearranging the order of its terms can make a conditionally convergent series converge to a different limit by considering the series $\sum(-1)^{n+1} n^{-1}=\ln 2=0.693$. Rearrange the series as
$$
S=\frac{1}{1}+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+\frac{1}{13}+\cdots
$$
and group each set of three successive terms. Show that the series can then be written
$$
\sum_{m=1}^{\infty} \frac{8 m-3}{2 m(4 m-3)(4 m-1)}
$$
which is convergent (by comparison with $\sum n^{-2}$ ) and contains only positive terms. Evaluate the first of these and hence deduce that $S$ is not equal to $\ln 2$.

Derrick Danso

Numerade Educator

Problem 18

Illustrate result (iv) of section $4.4$ about Cauchy products by considering the double summation
$$
S=\sum_{n=1}^{\infty} \sum_{r=1}^{n} \frac{1}{r^{2}(n+1-r)^{3}}
$$
By examining the points in the $n r$-plane over which the double summation is to be carried out, show that $S$ can be written as
$$
S=\sum_{n=r}^{\infty} \sum_{r=1}^{\infty} \frac{1}{r^{2}(n+1-r)^{3}}
$$
Deduce that $S \leq 3$

Eduard Sanchez

Numerade Educator

Problem 19

A Fabry-Pérot interferometer consists of two parallel heavily silvered glass plates; light enters normally to the plates, and undergoes repeated reflections between them, with a small transmitted fraction emerging at each reflection. Find the intensity $|B|^{2}$ of the emerging wave, where
$$
B=A(1-r) \sum_{n=0}^{\infty} r^{n} e^{i n \phi}
$$
with $r$ and $\phi$ real.

Derrick Danso

Numerade Educator

Problem 20

Identify the series
$$
\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^{2 n}}{(2 n-1) !}
$$
and then by integration and differentiation deduce the values $S$ of the following series,
(a) $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n^{2}}{(2 n) !}$,
(b) $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{(2 n+1) !}$,
(c) $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n \pi^{2 n}}{4^{n}(2 n-1) !}$,
(d) $\sum_{n=0}^{\infty} \frac{(-1)^{n}(n+1)}{(2 n) !}$.

William Semus

Numerade Educator

Problem 21

Starting from the Maclaurin series for $\cos x$, show that
$$
(\cos x)^{-2}=1+x^{2}+\frac{2 x^{4}}{3}+\cdots
$$
Deduce the first three terms in the Maclaurin series for $\tan x$.

Derrick Danso

Numerade Educator

Problem 22

Find the Maclaurin series for
(a) $\ln \left(\frac{1+x}{1-x}\right)$,
(b) $\left(x^{2}+4\right)^{-1}$
(c) $\sin ^{2} x$.

Daniel Pyda

Numerade Educator

Problem 23

If $f(x)=\sinh ^{-1} x$, and its $n$th derivative $f^{(n)}(x)$ is written as $P_{n}(x) /\left(1+x^{2}\right)^{n-1 / 2}$ where $P_{n}(x)$ is a polynomial (of order $\left.n-1\right)$, show that the $P_{n}(x)$ satisfy the recurrence relation
$$
P_{n+1}(x)=\left(1+x^{2}\right) P_{n}^{\prime}(x)-(2 n-1) x P_{n}(x)
$$
Hence generate the coefficients necessary to express $\sinh ^{-1} x$ as a Maclaurin series up to terms in $x^{5}$.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 24

Find the first three non-zero terms in the Maclaurin series for the following functions:
(a) $\left(x^{2}+9\right)^{-1 / 2}$,
(b) $\ln \left[(2+x)^{3}\right]$
(c) $\exp (\sin x)$,
(d) $\ln (\cos x)$,
(e) $\exp \left[-(x-a)^{-2}\right]$,
(f) $\tan ^{-1} x$.

Ahmed Kamel

Numerade Educator

Problem 25

By using the logarithmic series, prove that if $a$ and $b$ are positive and nearly equal then
$$
\ln \frac{a}{b} \simeq \frac{2(a-b)}{a+b}
$$
Show that the error in this approximation is about $2(a-b)^{3} /\left[3(a+b)^{3}\right]$.

Derrick Danso

Numerade Educator

Problem 26

Determine whether the following functions $f(x)$ are (i) continuous, and (ii) differentiable at $x=0:$
(a) $f(x)=\exp (-|x|)$;
(b) $f(x)=(1-\cos x) / x^{2}$ for $x \neq 0, f(0)=\frac{1}{2}$;
(c) $f(x)=x \sin (1 / x)$ for $x \neq 0, f(0)=0$
(d) $f(x)=\left[4-x^{2}\right]$, where $[y]$ denotes the integer part of $y$.

William Semus

Numerade Educator

Problem 27

Find the limit as $x \rightarrow 0$ of $\left[\sqrt{1+x^{m}}-\sqrt{1-x^{m}}\right] / x^{n}$, in which $m$ and $n$ are positive integers.

Derrick Danso

Numerade Educator

Problem 28

Evaluate the following limits:
(a) $\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sinh x}$,
(b) $\lim _{x \rightarrow 0} \frac{\tan x-\tanh x}{\sinh x-x}$,
(c) $\lim _{x \rightarrow 0} \frac{\tan x-x}{\cos x-1}$
(d) $\lim _{x \rightarrow 0}\left(\frac{\operatorname{cosec} x}{x^{3}}-\frac{\sinh x}{x^{5}}\right)$

Mir Afzal

Numerade Educator

Problem 29

Find the limits of the following functions:
(a) $\frac{x^{3}+x^{2}-5 x-2}{2 x^{3}-7 x^{2}+4 x+4}, \quad$ as $x \rightarrow 0, x \rightarrow \infty$ and $x \rightarrow 2$;
(b) $\frac{\sin x-x \cosh x}{\sinh x-x}, \quad$ as $x \rightarrow 0$
(c) $\int_{x}^{\pi / 2}\left(\frac{y \cos y-\sin y}{y^{2}}\right) d y, \quad$ as $x \rightarrow 0$.

Daniel Pyda

Numerade Educator

Problem 30

Use Taylor expansions to three terms to find approximations to (a) $\sqrt[4]{17}$, and (b) $\sqrt[3]{26}$.

Eduard Sanchez

Numerade Educator

Problem 31

Using a first-order Taylor expansion about $x=x_{0}$, show that a better approximation than $x_{0}$ to the solution of the equation
$$
f(x)=\sin x+\tan x=2
$$
is given by $x=x_{0}+h$, where
$$
h=\frac{2-f\left(x_{0}\right)}{\cos x_{0}+\sec ^{2} x_{0}}
$$
(a) Use this procedure twice to find the solution of $f(x)=2$ to six significant figures, given that it is close to $x=0.9$
(b) Use the result in (a) to deduce, to the same degree of accuracy, one solution of the quartic equation
$$
y^{4}-4 y^{3}+4 y^{2}+4 y-4=0
$$

Nick Johnson

Numerade Educator

Problem 32

Evaluate
$$
\lim _{x \rightarrow 0}\left[\frac{1}{x^{3}}\left(\operatorname{cosec} x-\frac{1}{x}-\frac{x}{6}\right)\right]
$$

Eduard Sanchez

Numerade Educator

Problem 33

In quantum theory, a system of oscillators, each of fundamental frequency $v$, interacting at temperature $T$ has an average energy $\bar{E}$ given by
$$
\bar{E}=\frac{\sum_{n=0}^{\infty} n h v e^{-n x}}{\sum_{n=0}^{\infty} e^{-n x}}
$$
where $x=h v / k T, h$ and $k$ being the Planck and Boltzmann constants respectively. Prove that both series converge, evaluate their sums, and show that at high temperatures $\bar{E} \approx k T$ whilst at low temperatures $\bar{E} \approx h v \exp (-h v / k T)$

Sam Stansfield

Numerade Educator

Problem 34

In a very simple model of a crystal, point-like atomic ions are regularly spaced along an infinite one-dimensional row with spacing $R$. Alternate ions carry equal and opposite charges $\pm e .$ The potential energy of the ith ion in the electric field due to the $j$ th ion is
$$
\frac{q_{i} q_{j}}{4 \pi \epsilon_{0} r_{i j}}
$$
where $q_{k}$ is the charge on the $k$ th ion and $r_{i j}$ is the distance between the $i$ th and $j$ th ions.

Write down a series giving the total contribution $V_{i}$ of the ith ion to the overall potential energy. Show that the series converges, and, if $V_{i}$ is written as
$$
V_{i}=\frac{\alpha e^{2}}{4 \pi \epsilon_{0} R}
$$
find a closed-form expression for $\alpha$, the Madelung constant for this (unrealistic) lattice.

Eduard Sanchez

Numerade Educator

Problem 35

One of the factors contributing to the high relative permittivity of water to static electric fields is the permanent electric dipole moment $p$ of the water molecule. In an external field $E$ the dipoles tend to line up with the field, but they do not do so completely because of thermal agitation at the temperature $T$ of the water. A classical (non-quantum) calculation using the Boltzmann distribution shows that the average polarisability per molecule $\alpha$ is given by
$$
\alpha=\frac{p}{E}\left(\operatorname{coth} x-x^{-1}\right)
$$
where $x=p E / k T$ and $k$ is the Boltzmann constant.
At ordinary temperatures, even with high field strengths $\left(10^{4} \mathrm{Vm}^{-1}\right.$ or more), $x \ll 1$. By making suitable series expansions of the hyperbolic functions involved, show that $\alpha=p^{2} / 3 k T$ to an accuracy of about one part in $15 x^{-2}$.

Derrick Danso

Numerade Educator

Problem 36

In quantum theory a certain method (the Born approximation) gives the (socalled) amplitude $f(\theta)$ for the scattering of a particle of mass $m$ through an angle $\theta$ by a uniform potential well of depth $V_{0}$ and radius $b$ (i.e. the potential energy of the particle is $-V_{0}$ within a sphere of radius $b$ and zero elsewhere) as
$$
f(\theta)=\frac{2 m V_{0}}{h^{2} K^{3}}(\sin K b-K b \cos K b)
$$
Here $\hbar$ is the Planck constant divided by $2 \pi$, the energy of the particle is $h^{2} k^{2} / 2 m$ and $K$ is $2 k \sin (\theta / 2)$

Use l'Hôpital's rule to evaluate the amplitude at low energies, i.e. when $k$ and hence $K$ tend to zero, and so determine the low-energy total cross-section.
(Note: the differential cross-section is given by $|f(\theta)|^{2}$ and the total cross-section by the integral of this over all solid angles, i.e. $2 \pi \int_{0}^{\lambda}|f(\theta)|^{2} \sin \theta d \theta .$ )

Eduard Sanchez

Numerade Educator