THEOREM 2a. If converges, then
This provides a convenient and useful test for divergence, since it is equivalent to the statement: If
an does not approach zero, then the series diverges. Note, however, particularly that the converse
of Theorem 2a is not true. The condition that an approach zero is necessary but not sufficient for the
convergence of the series. The harmonic series is an excellent example of a series whose nth
term goes to zero but that diverges (see Example 11 above). The series diverges because
not zero; the series does not converge (as will be shown shortly) even though
THEOREM 2b. A finite number of terms may be added to or deleted from a series without affecting
its convergence or divergence; thus
(where m is any positive integer) both converge or both diverge. (Note that the sums most likely will
differ.)
THEOREM 2c. The terms of a series may be multiplied by a nonzero constant without affecting the
convergence or divergence; thus
both converge or both diverge. (Again, the sums will usually differ.)
THEOREM 2d. If both converge, so does
THEOREM 2e. If the terms of a convergent series are regrouped, the new series converges.
B3. Tests for Convergence of Infinite Series.
THE nth TERM TEST
If diverges.
NOTE: When working with series, it’s a good idea to start by checking the nth Term Test. If the
terms don’t approach 0, the series cannot converge. This is often the quickest and easiest way to
identify a divergent series.
(Because this is the contrapositive of Theorem 2a, it’s always true. But beware of the converse!
Seeing that the terms do approach 0 does not guarantee that the series must converge. It just means
that you need to try other tests.)
EXAMPLE 12
Does converge or diverge?
SOLUTION: Since the series diverges by the nth Term Test.THE GEOMETRIC SERIES TEST
A geometric series converges if and only if |r| < 1.
If |r| < 1, the sum is
The series cannot converge unless it passes the nth Term Test; only if |r| < 1. As noted