Page 100 - Calculus for the Clueless, Calc II

Note 2

There are three possibilities for an alternating series: it diverges, converges conditionally, or converges
absolutely.

Let us look at three alternating series.

Example 31—

Let us look at . This series converges conditionally since (1) the terms go to zero, but (2)
using the limit comparison test with the harmonic series, the positive series behaves as the harmonic series and
diverges.

Example 32—

What about the series ? This series converges absolutely by comparing to the p 2 series.

Example 33—

diverges since the terms don't go to 0.

Definition

Region of convergence—We have an infinite series whose terms are functions of x. The set of all points x for
which the series converges is called the region of convergence.

Now let's get back to the series with x's in them. Series of this type are usually done with the ratio test. This is
to find the region of convergence. Then, you will test both the left and right end points. There are three tests in
all.

Example 34—

Using the ratio test,

So the region of convergence is |x| < 1 or -1 < x < 1.

Let us test both 1 and -1 by substituting those values into the original series. x = 1 gives us or
, the harmonic series that diverges. x = -1 gives us , the alternating harmonic series, or
rather the negative of the alternating harmonic series, since the first term is negative. We know this converges.
Therefore, the region of convergence is -1 < x < 1.

Important Note

When you test the end points, anything is possible. Both ends could converge, both could diverge, the left could
converge and not the right, or the right could converge but not the left.

Example 35—