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)