Alternating Series

Alternating Series:
is alternating if its terms be a series. Then Let
are alternating positive and negative.

= either So,
Or
is positive. Where

Example:
The following are alternating series:
. (1)
. (2)


The alternating Series Test:
Theorem (Alternating Series Test):
Let
be an alternating series with
for all n. Then this series converges if (1)
. (2)

Example:
Determine which of the following alternating series converge
. (a)
. (b)
. (c)



Solution:
, clearly: (a) Here
, and (1)
. = (2)
Hence we deduce from the alternating series test that the given series converges .
Converges . i.e.,
, clearly: (b) Here
, and (1)
. = (2)
Hence we deduce from the alternating series test that the given series converges .
Converges . i.e.,

, clearly: (c) Here
. =
Hence we deduce from the alternating series test that the given series diverges .
diverges . i.e.,

Exercise:
Which of the following alternating series converge:
. (a)
. (b)
. (c)