Chapter Questions
Find (a) the general term and (b) the recurrence relation for the sequences:$1,4,7,10, \ldots$
$1,3,9,27, \ldots$
$1,-\frac{1}{5}, \frac{1}{25},-\frac{1}{125}, \ldots$
Find the first 6 terms of the sequences:$u_{r+1}=u_{r}+\frac{1}{2} ; \quad u_{1}=0$
$v_{n}=\left(\frac{2}{3}\right)^{n} ; \quad n=0,1,2, \ldots$
$u_{x}=\frac{1}{x(x+2)} ; \quad x=1,2,3, \ldots$
$w_{n+1}=\frac{w_{n}}{n} ; \quad w_{1}=1$
$u_{n+2}=u_{n+1}+2 u_{n} ; \quad u_{0}=1, u_{1}=3$
$u_{n+2}=3 u_{n+1}-2 u_{n} ; \quad u_{0}=1, u_{1}=1 / 2$
$u_{n+2}=3 u_{n+1}-2 u_{n} ; \quad u_{0}=u_{1}$
Find the limit $r \rightarrow \infty$ for:$\frac{1}{3^{r}}$
$2^{r}$
$\frac{1}{r+2}$
$\frac{r}{r+2}$
$\frac{r}{r^{2}+r+1}$
$\frac{3 r^{2}+3 r+1}{5 r^{2}-6 r-1}$
Find the limit of the sequence $\left\{u_{n+1} / u_{n}\right\}$ for $u_{n+2}=u_{n+1}+2 u_{n} ; \quad u_{0}=1, \quad u_{1}=3$ (see Exercise 8).
Find the sum of (i) the first $n$ terms, (ii) the first 10 terms:$1+5+9+13+\cdots$
$3-2-7-12-\cdots$
$1+3+9+27+\cdots$
$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$
Find the sum of the first $n$ terms:$x^{3}+x^{5}+x^{7}+\cdots$
$x+2 x^{2}+4 x^{3}+\cdots$
Use equation (7.11) to expand in powers of $x$ :$(1+x)^{5}$
$(1+x)^{7}$
Calculate the binomial coefficients $\left(\begin{array}{l}n \\ r\end{array}\right), r=0,1, \ldots, n$, for $n=3$
$n=4$
$n=7$
Use equation (7.13) or (7.14) to expand in powers of $x$ :$(1-x)^{3}$
$(1+3 x)^{4}$
$(1-4 x)^{5}$
$(3-2 x)^{4}$
$(3+x)^{6}$
(i) Calculate the distinct trinomial coefficients $\frac{4 !}{n_{1} ! n_{2} ! n_{3} !}$. (ii) Use the coefficients to expand $(a+b+c)^{4} .$
(i) Calculate the distinct coefficients $\frac{3 !}{n_{1} ! n_{2} ! n_{3} ! n_{4} !}$. (ii) Use the coefficients to expand
Find $\sum_{n=1}^{10} \frac{1}{n(n+1)}$.
(i) Verify that $\frac{1}{r(r+2)}=\frac{1}{2}\left(\frac{1}{r}-\frac{1}{r+2}\right)$, then (ii) find the sum of the series $\sum_{r=1}^{n} \frac{1}{r(r+2)}$.
(i) Express $\frac{1}{r(r+1)(r+2)}$ in partial fractions, then (ii) show that$$\sum_{r=1}^{n} \frac{1}{r(r+1)(r+2)}=\frac{1}{4}-\frac{1}{2(n+1)(n+2)}$$
(i) Verify that $(1+r)^{3}-r^{3}=3 r^{2}+3 r+1$, then (ii) show that $\sum_{r=1}^{n} r^{2}=\frac{1}{6} n(n+1)(2 n+1)$
(i) Expand $(1+r)^{6}-r^{6}$, then (ii) use the series in Table $7.1$ to find the sum of the series $\sum_{r=1}^{n} r^{5}$.
(i) Expand in powers of $x$ to terms in $x^{6}$. (ii) Find the values of $x$ for which the series converge:$\frac{1}{1-3 x}$
$\frac{1}{1+5 x^{2}}$
$\frac{1}{2+x}$
(i) Use the geometric series to express the number $1 /\left(10^{6}-1\right)$ as a decimal fraction.(ii) Show that the decimal representation of $1 / 7$ can be written as $142857 /\left(10^{6}-1\right)$ (see Section 1.4).
The vibrational partition function of a harmonic oscillator is given by the series$$q_{\mathrm{v}}=\sum_{n=0}^{\infty} e^{-n \theta_{\mathrm{v}} / T}$$where $\theta_{\mathrm{v}}=h v_{\mathrm{c}} / k$ is the vibrational temperature. Confirm that the series is a convergent geometric series, and find its sum.
Examine the following series for convergence by Comparison test (use $\ln n<n$ ):D'Alembert ratio test:Cauchy integral test:$\sum_{n=2}^{\infty} \frac{1}{\ln n}$
$\sum_{r=1}^{\infty} \frac{\ln r}{r^{3}}$
$\sum_{s=0}^{\infty} \frac{s^{a}}{(s+1) !}$
$\sum_{r=1}^{\infty} \frac{1}{r^{a}}$
$\sum_{n=2}^{\infty} \frac{1}{n \ln n}$
Find the radius of convergence of each of the following series:$\sum_{m=0}^{\infty} \frac{x^{m}}{4^{m}}$
$\sum_{r=0}^{\infty}(-1)^{r} x^{2 r}$
$\sum_{n=1}^{\infty} n x^{n}$
$\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}}$
$\sum_{m=1}^{\infty} m^{m} x^{m}$
$\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{3^{n}}$
Write down the first 5 terms of the MacLaurin series of the following functions:$(1+x)^{1 / 3}$
$\frac{1}{1+x^{2}}$
$(1-x)^{-1 / 2}$
$\frac{1}{3+x}$
$\sin 2 x^{2}$
$\frac{\ln (1-2 x)+2 x}{x^{2}}$
$e^{-3 x}$
$\frac{e^{x^{2}}-1}{x}$
A body with rest mass $m_{0}$ and speed $v$ has relativistic energy$$E=m c^{2}=\frac{m_{0} c^{2}}{\sqrt{1-v^{2} / c^{2}}}$$and kinetic energy $T=E-m_{0} c^{2}$. Express $T$ as a power series in $v$ and show that the series reduces to the nonrelativistic kinetic energy in the limit $v / c \rightarrow 0$.
The equation of state of a gas can be expressed in terms of the series$$p V=n R T \sum_{i=0}^{\infty} B_{i}(T)\left(\frac{n}{V}\right)^{i}$$where the $B_{i}$ are called virial coefficients. Find the first three coefficients for(i) the van der Waals equation, $\left(p+\frac{n^{2} a}{V^{2}}\right)(V-n b)=n R T$(ii) the Dieterici equation, $p(V-n b)=n R T e^{-a n / R T V}$
(i) Expand each of the following functions as a Taylor series about the given point, and (ii) find the values of $x$ for which the series converges:$\frac{1}{x}, 1$
$e^{x}, 2$
$\sin x, \pi / 2$
$\ln x, 2$
(i) Find the MacLaurin expansion of the function $(8+x)^{1 / 3}$ up to terms in $x^{4}$. (ii) Use this expansion to find an approximate value of $\sqrt[3]{9}$. (iii) Use this value and Taylor's theorem for the remainder to compute upper and lower bounds to the value of $\sqrt[3]{9}$.
Find the limits:$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$
$\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}}$
$\lim _{x \rightarrow 0} \frac{e^{x}+e^{-x}-2}{\cos x-1}$
$\lim _{x \rightarrow 1} \frac{\ln x}{x^{2}-1}$
The energy density of black-body radiation at temperature $T$ is given by the Planck formula$$\rho(\lambda)=\frac{8 \pi h c}{\lambda^{5}}\left[e^{h c / \lambda k T}-1\right]^{-1}$$where $\lambda$ is the wavelength. Show that the formula reduces to the classical Rayleigh-Jeans law $\rho=8 \pi k T / \lambda^{4}$ (i) for long wavelengths $(\lambda \rightarrow \infty)$, (ii) if Planck's constant is set to zero $(h \rightarrow 0)$.
Find the Cauchy product of the power series expansions of $\sin x$ and $\cos x$, and show that it is equal to $\frac{1}{2} \sin 2 x$.
Differentiate the power series expansion of $\sin x$ and show that the result is $\cos x$.
Integrate the power series expansion of $\sin x$ and show that the result is $C-\cos x$, where $C$ is a constant.