1. Section 7: Positive-Term Series

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

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

4. Ex 1: Test the series for convergence or divergence.

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

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

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

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

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

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

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

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

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

14. Section 7 WS #1 – 33 EOO, 34 – 40 all