Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.8 EXERCISES

4.20 Identify the series

∑∞

n=1

(−1)n+1x^2 n

(2n−1)!

,

and then, by integration and differentiation, deduce the valuesSof the following

series:

(a)

∑∞

n=1

(−1)n+1n^2

(2n)!

,(b)

∑∞

n=1

(−1)n+1n

(2n+1)!

,

(c)

∑∞

n=1

(−1)n+1nπ^2 n

4 n(2n−1)!

,(d)

∑∞

n=0

(−1)n(n+1)

(2n)!

.

4.21 Starting from the Maclaurin series for cosx, show that

(cosx)−^2 =1+x^2 +

2 x^4

3

+···.

Deduce the first three terms in the Maclaurin series for tanx.
4.22 Find the Maclaurin series for:

(a) ln

(

1+x

1 −x

)

, (b) (x^2 +4)−^1 , (c) sin^2 x.

4.23 Writing thenth derivative off(x)=sinh−^1 xas

f(n)(x)=

Pn(x)

(1 +x^2 )n−^1 /^2

,

wherePn(x) is a polynomial (of ordern−1), show that thePn(x)satisfythe

recurrence relation

Pn+1(x)=(1+x^2 )Pn′(x)−(2n−1)xPn(x).

Hence generate the coefficients necessary to express sinh−^1 xas a Maclaurin series
up to terms inx^5.
4.24 Find the first three non-zero terms in the Maclaurin series for the following
functions:

(a) (x^2 +9)−^1 /^2 , (b) ln[(2 +x)^3 ], (c) exp(sinx),

(d) ln(cosx), (e) exp[−(x−a)−^2 ], (f) tan−^1 x.

4.25 By using the logarithmic series, prove that ifaandbare positive and nearly
equal then

ln

a

b



2(a−b)

a+b

.

Show that the error in this approximation is about 2(a−b)^3 /[3(a+b)^3 ].
4.26 Determine whether the following functionsf(x) are (i) continuous, and (ii)
differentiable atx=0:

(a) f(x)=exp(−|x|);

(b)f(x)=(1−cosx)/x^2 forx=0,f(0) =^12 ;

(c) f(x)=xsin(1/x)forx=0,f(0) = 0;

(d)f(x)=[4−x^2 ], where [y] denotes the integer part ofy.

4.27 Find the limit asx→0of[

√

1+xm−

√

1 −xm]/xn,inwhichmandnare positive
integers.
4.28 Evaluate the following limits: