Advanced High-School Mathematics

274 CHAPTER 5 Series and Differential Equations

seriesa+ar+ar^2 +··· converges (to

a

1 −r

). In this test we do not

need to assume that the series consists only of non-negative terms.

The Ratio Test Let

∑∞

n=0

anbe an infinite series. Assume that

nlim→∞

an+1

an

=R.

Then

(i) if|R|< 1 , then

∑∞

n=0

anconverges;

(ii) if|R|> 1 , then

∑∞

n=0

andiverges;

(iii) if|R|= 1, then this test is inconclusive.

The reasoning behind the above is simple. First of all, in case (i) we

see that

∑∞

n=0

anis asymptotically a geometric series with ratio|R|< 1

and hence converges (but we still probably won’t know what the series

converges to). In case (ii) then

∑∞

n=0

anwill diverge since asymptotically

each term is R times the previous one, which certainly implies that

nlim→∞an^6 = 0, preventing convergence. Note that in the two cases

∑∞

n=1

1

n

and

∑∞

n=1

1

n^2

we have limn→∞

an+1

an

= 1,^10 which is why this case is inclusive.

We turn again to some examples.

Example 7. Consider the series

∑∞

n=1

(n+ 1)^3

n!

. We have

(^10) Indeed, we have in the first case
nlim→∞ana+1
n
= limn→∞
Ä 1
n+1
ä
( 1
n
) = limn→∞nn+ 1= 1,
in the first case, and that
nlim→∞aan+1n = limn→∞
Ä 1
(n+1)^2
ä
( 1
n^2
) = limn→∞nn+ 1= 1,
in the second case, despite the fact that the first series diverges and the second series converges.