Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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:

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