Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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

.