Barrons AP Calculus

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.) ABSOLUTE CONVERGENCE AND CONDITIONAL
CONVERGENCE

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.

Conditional convergence

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.