Page 95 - Calculus for the Clueless, Calc II

It is necessary that for to converge.

Note 1

If a k does not go to 0, diverges.

Note 2

If a k does go to O, and that is all we know, we know nothing.

Example 14—

Tell whether converges.

k/(k + 1) goes to 1. Therefore diverges.

Example 15—

The harmonic series

Since 1/k goes to 0, we don't know if this series converges or diverges. We shall shortly show that the harmonic
series diverges.

Example 16—

The p 2 series
Since 1/k goes to 0, again we can't tell. Shortly we shall show that the P 2 series converges.
2

Test 2

Given a k and a k > 0, a k goes to 0 for k big enough. Suppose we have a continuous function f(x) such that f(k) =
a k. Then and dx either both converge or both diverge.

This theorem is easily explained by examples.

Example 17—

Tell whether converges or diverges.

The improper integral associated with is dx. Letting u = -x 2, du =2x dx.

Since the improper integral converges, so does the infinite series.

Note 1

The value of the improper integral is not the value of the infinite series. But we can say the following: If the
integral and the series together converge, then dx.