Barrons AP Calculus - David Bock

THE ALTERNATING SERIES TEST

An alternating series converges if:
(1) an + 1 < an for all n, and
(2)
EXAMPLE 26
Does the series converge or diverge?
SOLUTION: The alternating harmonic series converges, since
(1) for all n and
(2)

EXAMPLE 27

Does the series converge or diverge?

SOLUTION: The series diverges, since we see that is 1, not 0. (By the

nth Term Test, if an does not approach 0, then does not converge.)

DEFINITION

Absolute

convergence

A series with mixed signs is said to converge absolutely (or to be absolutely convergent) if the
series obtained by taking the absolute values of its terms converges; that is, converges absolutely
if converges.
A series that converges but not absolutely is said to converge conditionally (or to be
conditionally convergent). The alternating harmonic series converges conditionally since it
converges, but does not converge absolutely. (The harmonic series diverges.)
When asked to determine whether an alternating series is absolutely convergent, conditionally
convergent, or divergent, it is often advisable to first consider the series of absolute values. Check
first for divergence, using the nth Term Test. If that test shows that the series may converge,
investigate further, using the tests for nonnegative series. If you find that the series of absolute values
converges, then the alternating series is absolutely convergent. If, however, you find that the series of
absolute values diverges, then you’ll need to use the Alternating Series Test to see whether the series
is conditionally convergent.

EXAMPLE 28

Determine whether converges absolutely, converges conditionally, or diverges.

SOLUTION: We see that not 0, so by the nth Term Test the series is