194 Chapter4fails, unless Kn :::; 0 for n > N, in which case the series diverges by the
first form of Kummer's test, provided that 2::(1/ Dn) is divergent.
Third form. If liminfn--+oo Kn > 0, then 2:: \an\ converges. If
limsupn--+ooKn < 0 and 2::(1/Dn) diverges, then L:\an\ diverges.
For proofs of the preceding tests, the reader is referred to Smail [10].
( d) Ratio test (D'Alembert, 1768). This is a particular case of Kummer's
test obtained by choosing Dn = ·1 for every n. The corresponding three
forms of the test (the second of which is the most commonly used) can be
expressed as follows:
First form. If \an+i\/\an\ :=:; r < 1 for n> N, then L:\an\ is convergent,
and if \an+1\/\an\;::: 1,.then L:an is divergent.
Note Here we may assert the divergence of 2:: an, rather than that of
L \an [ since the assumption implies that \an+1 \ ;::: \an\ ;::: · · · ;::: \aN+1 [ > 0,
so an f> 0. The same observation is valid for the other two forms.
Second form. Suppose that limn--+oo \an+1 \/\an\ = L (L finite or infinite).
Then if L < 1, the series I: \an[ converges, and if L > 1, the series diverges.
If L = 1, the test fails, unless \an+1\/\an\ ;::: 1 for n > N, in which case
2:: an diverges.Third form. If limsupn--+oo \an+1\/[an\ < 1, then 2:: \an\ converges, and
ifliminfn--+oo[an+i[/[an[ > 1, then L:an diverges.
( e) Raabe-Duhamel test. This test may be used when limn--+oo [an+1 [/\an\ = 1 and \an+i\/\an\ ;::: 1 does not hold from some value of non, i.e.,
in case the second form of the ratio test fails.
Raabe's test is also a particular case of Kummer's test, obtained by
letting Dn = n. Using the second form of that test, we get: If
lim n ( ~ - 1) = L
n--+oo \an+i \
then 2:: \an\ converges if L > 1, and diverges if L < 1.
If L = 1, the test fails, unless
n(~ -1) <1
[an+1[ -for n > N, in which case 2:: \an\ diverges, since the preceding inequality
can also be written as
\an+1\ > l/(n + 1)
\an[ - l/n
so 2:: \an[ diverges by comparison with 2::(1/n).
(f) Many other particular tests can be derived from Kummer's rule.
For instance, for the case in which Raabe's test fails, we may proceed by