Page 99 - Calculus for the Clueless, Calc II

To show the third part (r = 1), we would again use the harmonic and P2 series. Let us give examples for the first
two parts (r > 1 and r < 1).

Example 28—

k
k l/k
Take (3 /k ) = 3/k. . So the series converges.
Example 29—

k
Take (2 /k ) = 2/k . . Since 2 > 1, the series
2/k
2 l/
diverges.
Up to this time, we have dealt exclusively with positive terms. Now we will deal with infinite series that have
terms that alternate from positive to negative. We will assume the first term is positive. The notation will be as
follows: alternating series , where all a k are positive.

TEST 7

Given an alternating series where (A) 0 < A k+1 < a k, k = 1, 2, 3, 4, .... and (b) , the series converges
to S and S < a 1

In clearer English, the only thing you must do to show an alternating series converges is to show the terms go to
zero. (If only all series were that easy?)
Example 30—

Alternating harmonic

converges since the terms go to 0.

Definition

Absolutely convergent—A series converges absolutely if converges.

Note

If a series converges absolutely, it converges.

Definition

Conditionally convergent—A series converges conditionally if it converges but diverges.

Note 1

If we have an alternating series and want to show that it converges conditionally, we only have to show its
terms go to zero. To find out whether it is absolutely convergent, we must use some other test.