Page 97 - Calculus for the Clueless, Calc II

Example 21—

Examine .

4
4
1/(4 + k ) < 1/k . converges by the previous example. Since the given series is smaller termwise than
a convergent series, it must converge by the comparison test.

Example 22—

Examine .

a divergent series (the harmonic). Since the given series is larger term-
wise than a divergent series, the given series must diverge.

Test 4

This is the limit comparison test. Given and . If , where r is any
positive number, both series converge, or both diverge.

Example 23—

Let us compare this series with .

4
Divide top and bottom by k .

Since the limit is a positive number, both series do the same thing. Since converges, so does

Test 5 (Ratio Test)

Given a k > 0, If r > 1, it diverges. If r < 1, it converges. If r = 1, use another test.

Example 24—

Examine .

The series converges.