Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.3 CONVERGENCE OF INFINITE SERIES

Ratio comparison test

As its name suggests, the ratio comparison test is a combination of the ratio and

comparison tests. Let us consider the two series

∑

unand

∑

vnand assume that

we know the latter to be convergent. It may be shown that if

un+1

un

≤

vn+1

vn

for allngreater than some fixed valueNthen

∑

unis also convergent.

Similarly, if

un+1

un

≥

vn+1

vn

for all sufficiently largen,and

∑

vndiverges then

∑

unalso diverges.

Determine whether the following series converges:

∑∞

n=1

1

(n!)^2

=1+

1

22

+

1

62

+···.

In this case the ratio of successive terms, asntends to infinity, is given by

R= lim

n→∞

[

n!

(n+1)!

] 2

= lim

n→∞

(

1

n+1

) 2

,

which is less than the ratio seen in (4.11). Hence, by the ratio comparison test, the series
converges. (It is clear that this series could also be found to be convergent using the ratio
test.)

Quotient test

The quotient test may also be considered as a combination of the ratio and

comparison tests. Let us again consider the two series

∑

unand

∑

vn, and define

ρas the limit

ρ= lim

n→∞

(

un

vn

)

. (4.12)

Then, it can be shown that:

(i) ifρ= 0 but is finite then

∑

unand

∑

vneither both converge or both

diverge;

(ii) ifρ= 0 and

∑

vnconverges then

∑

unconverges;

(iii) ifρ=∞and

∑

vndiverges then

∑

undiverges.