B2. Theorems About Convergence or Divergence of Infinite Series
The following theorems are important.
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 and both converge, so does .
THEOREM 2e. If the terms of a convergent series are regrouped, the new series
converges.
BC ONLYB3. 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