Chapter 1: Infinite Series, Power Series

Define a Function$\mathit{h}\left(x\right)\mathbf{=}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\mathit{f}\left(x+2k\pi \right)$, assuming that the series converges to a function satisfying Dirichlet conditions (Section 6). Verify that h(x)does have Period$\mathbf{2}\mathit{\pi }$.

(a) Expand h(x)in an exponential Fourier series$\mathit{h}\left(x\right)\mathbf{=}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathit{c}}_{\mathbf{n}}{\mathit{e}}^{\mathbf{i}\mathbf{n}\mathbf{x}}$;show that ${\mathit{c}}_{\mathbf{n}}\mathbf{=}\mathit{g}\left(n\right)$where $\mathit{g}\left(\alpha \right)$is the Fourier transform of f(x). Hint: Write ${\mathit{c}}_{\mathbf{n}}$as anIntegral from 0to $\mathbf{2}\mathit{\pi }$and make the change of variable $\mathit{u}\mathbf{=}\mathit{x}\mathbf{+}\mathbf{2}\mathit{k}\mathit{\pi }$. Note that${\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{i}\mathbf{n}\mathbf{k}\mathbf{\pi }}\mathbf{=}\mathbf{1}$, and the sum onkgives a single integral from $\mathbf{-}\mathbf{\infty }\mathbf{}\mathit{t}\mathit{o}\mathbf{}\mathbf{\infty }$

(b) Let x=0in (a) to get Poisson’s summation formula $\mathbf{\sum }_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\mathbf{f}\left(2k\pi \right)\mathbf{=}\mathbf{\sum }_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\mathit{g}\left(n\right)$

This result has many applications; for example: statistical mechanics, communicationtheory, theory of optical instruments, scattering of light in a liquid and so on (See Problem 22).

Test for convergence:$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{n}\mathbf{l}\mathbf{n}{\mathbf{\left(}\mathbf{n}\mathbf{\right)}}^{\mathbf{3}}}$

For the following series, write formulas for the sequences ${\mathit{a}}_{\mathbf{n}}\mathbf{,}{\mathit{S}}_{\mathbf{n}}\mathbf{,}\mathbf{\text{and}}{\mathit{R}}_{\mathbf{n}}$, and find the limits of the sequences as $\mathit{n}\mathbf{\to }\mathbf{\infty }$ (if the limits exist).

$\frac{\mathbf{3}}{\mathbf{1}\mathbf{·}\mathbf{2}}\mathbf{-}\frac{\mathbf{5}}{\mathbf{2}\mathbf{·}\mathbf{3}}\mathbf{+}\frac{\mathbf{7}}{\mathbf{3}\mathbf{·}\mathbf{4}}\mathbf{-}\frac{\mathbf{9}}{\mathbf{4}\mathbf{·}\mathbf{5}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

For the force field $\mathit{F}\mathbf{=}\left(y+z\right)\mathit{i}\mathbf{-}\left(x+z\right)\mathit{j}\mathbf{+}\left(x+y\right)\mathit{k}$
, find the work done in moving a particle around each of the following closed curves:

(a) the circle ${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$ in the (x,y) plane, taken counterclockwise;

(b) the circle ${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$ in the (x,z) plane, taken counterclockwise;

(c) the curve starting from the origin and going successively along the x axis to (1,0,0) , parallel to the z axis to (1,0,1) , parallel to the (y,z) plane to (1,1,1) , and back to the origin along x = y = z

(d) from the origin to $\left(0,0,2\pi \right)$ on the curve

$\mathit{x}\mathbf{=}\mathbf{1}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathit{t}$ ,

$\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$ ,

$\mathit{z}\mathbf{=}\mathit{t}$

and back to the origin along the z axis.

$\frac{\mathbf{1}}{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{{\mathbf{d}}^{\mathbf{10}}}{{\mathbf{dx}}^{\mathbf{10}}}\left(x^8{\tan }^{2}x\right)\mathbf{}\mathbf{at}\mathbf{}\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}$

Test the following series for convergence

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{{}^{\mathbf{\infty }}}\frac{\mathbf{(}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{n}}\mathbf{n}}{\mathbf{1}\mathbf{+}{\mathbf{n}}^{\mathbf{2}}}$

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.

7.$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{\mathbf{1}}{\mathbf{n}\mathbf{ }\mathbf{lnn}}$

Use equation (1.8) to find the fraction that are equivalent to following repeating decimals.

7. 0.1851185…

Show that$\frac{\mathbf{2}}{\sqrt{\mathbf{4}\mathbf{-}\mathbf{x}}}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{8}}\mathit{x}$with an error less than$\frac{\mathbf{1}}{\mathbf{32}}$for 0<x<1. Hint: Let x=4y , and use the theorem (14.4) .