Page 269 - Calculus Workbook For Dummies

253
Chapter 13: Infinite Series: Welcome to the Outer Limits

3
3
19. ! n n 20. ! ! n n
=
n 1 ! n n 1 4
=
Solve It Solve It

He Loves Me, He Loves Me Not:

Alternating Series

n
Alternating series look just like any other series except that they contain an extra - 1h
^
n 1
+
or - 1h . This extra term causes the terms of the series to alternate between positive
^
and negative.
An alternating series converges if two conditions are met:
1. Its nth term converges to zero.
2. Its terms are non-increasing — in other words, each term is either smaller than
or the same as its predecessor (ignoring the minus sign).

For the problems in this section, determine whether the series converges or diverges.
If it converges, determine whether the convergence is absolute or conditional.

If you take a convergent alternating series and make all the terms positive and it still
converges, then the alternating series is said to converge absolutely. If, on the other
hand, the series of positive terms diverges, then the alternating series converges
conditionally.