Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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4.8 EXERCISES


4.8 TheN+ 1 complex numbersωmare given byωm=exp(2πim/N), form=
0 , 1 , 2 ,... ,N.


(a) Evaluate the following:

(i)

∑N


m=0

ωm, (ii)

∑N


m=0

ωm^2 , (iii)

∑N


m=0

ωmxm.

(b) Use these results to evaluate:

(i)

∑N


m=0

[


cos

(


2 πm
N

)


−cos

(


4 πm
N

)]


, (ii)

∑^3


m=0

2 msin

(


2 πm
3

)


.


4.9 Prove that


cosθ+cos(θ+α)+···+cos(θ+nα)=

sin^12 (n+1)α
sin^12 α

cos(θ+^12 nα).

4.10 Determine whether the following series converge (θandpare positive real
numbers):


(a)

∑∞


n=1

2sinnθ
n(n+1)

, (b)

∑∞


n=1

2


n^2

, (c)

∑∞


n=1

1


2 n^1 /^2

,


(d)

∑∞


n=2

(−1)n(n^2 +1)^1 /^2
nlnn

, (e)

∑∞


n=1

np
n!

.


4.11 Find the real values ofxfor which the following series are convergent:


(a)

∑∞


n=1

xn
n+1

, (b)

∑∞


n=1

(sinx)n, (c)

∑∞


n=1

nx,

(d)

∑∞


n=1

enx, (e)

∑∞


n=2

(lnn)x.

4.12 Determine whether the following series are convergent:


(a)

∑∞


n=1

n^1 /^2
(n+1)^1 /^2

, (b)

∑∞


n=1

n^2
n!

, (c)

∑∞


n=1

(lnn)n
nn/^2

, (d)

∑∞


n=1

nn
n!

.


4.13 Determine whether the following series are absolutely convergent, convergent or
oscillatory:


(a)

∑∞


n=1

(−1)n
n^5 /^2

, (b)

∑∞


n=1

(−1)n(2n+1)
n

, (c)

∑∞


n=0

(−1)n|x|n
n!

,


(d)

∑∞


n=0

(−1)n
n^2 +3n+2

, (e)

∑∞


n=1

(−1)n 2 n
n^1 /^2

.


4.14 Obtain the positive values ofxfor which the following series converges:


∑∞

n=1

xn/^2 e−n
n

.

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