The Divergence and Integral Tests

Theorem 8.8 Divergence Test

If convergences, then   If the limit does not equal 0, then the series diverges.

Theorem 8.9 The HarmonicSeries

The Harmonic Series diverges even though the terms approach zero

Theorem 8.10 Integral Test

 Suppose f is a continuous, positive, and decreasing function for , and let for k= 1, 2, 3, 4.... Then and either both converge or both diverge. In the case of convergence, the value of the integral is not equal to the value of the series

Theorem 8.11 Convergence of p-Series

The p-series converges for and diverges for

Properties of Convergent Series

Suppose converges to A and converges to b. Then A) B)


Geometric proof of integral test