Section 7: Positive-Term Series

The Integral Test Suppose f is a continuous, positive, decreasing function and let an = f(n):

The Integral Test * If is convergent, Then is convergent. * If is divergent, Then is divergent.

Ex 1: Test the series for convergence or divergence.

P-Series Test: * The p-series is convergent if p >1 and divergent if p  1.

Ex 2: Determine whether the series converges or diverges. A) B)

The Direct Comparison Test Suppose an  0, bn  0, and an ≤ bn for all n:

The Direct Comparison Test * If is convergent, Then is convergent. * If is divergent, Then is divergent.

Ex 3: Determine whether the series converges or diverges. A) B)

Ex 3: Determine whether the series converges or diverges. C)

The Limit Comparison Test Suppose an  0 and bn  0 for all n:

The Limit Comparison Test * If exists and is both positive and finite, then and either both converge or both diverge.

Ex 4: Determine whether the series converges or diverges. A) B)

Section 7 WS #1 – 33 EOO, 34 – 40 all