Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.3 CONVERGENCE OF INFINITE SERIES

Using the integral test, we consider

lim

N→∞

∫N

1

xp

dx= lim

N→∞

(

N^1 −p

1 −p

)

,

and it is obvious that the limit tends to zero forp>1andto∞forp≤1.

Cauchy’s root test

Cauchy’s root test may be useful in testing for convergence, especially if thenth

terms of the series contains annth power. If we define the limit

ρ= lim

n→∞

(un)^1 /n,

then it may be proved that the series

∑

unconverges ifρ<1. Ifρ>1 then the

series diverges. Its behaviour is undetermined ifρ=1.

Determine whether the following series converges:

∑∞

n=1

(

1

n

)n

=1+

1

4

+

1

27

+···.

Using Cauchy’s root test, we find

ρ= lim

n→∞

(

1

n

)

=0,

and hence the series converges.

Grouping terms

We now consider the Riemann zeta series, mentioned above, with an alternative

proof of its convergence that uses the method of grouping terms. In general there

are better ways of determining convergence, but the grouping method may be

used if it is not immediately obvious how to approach a problem by a better

method.

First consider the case wherep>1, and group the terms in the series as

follows:

SN=

1

1 p

+

(

1

2 p

+

1

3 p

)

+

(

1

4 p

+···+

1

7 p

)

+···.

Now we can see that each bracket of this series is less than each term of the

geometric series

SN=

1

1 p

+

2

2 p

+

4

4 p

+···.

This geometric series has common ratior=

( 1

2

)p− 1

;sincep>1, it follows that

r<1 and that the geometric series converges. Then the comparison test shows

that the Riemann zeta series also converges forp>1.