1. Section 9.3 Convergence of Sequences and Series

2. Consider a general series The partial sums for a sequence, or string of numbers written The sequence converges if Otherwise the sequence diverges –We have already looked at geometric series

3. Examples

4. Convergence of Series If S n is convergent, so, then the series is convergent and the sum is S. Otherwise it diverges.

5. Theorem Note: It is not enough to show that to show a series is convergent Let’s look at the following example

6. Comparison of Series and Integrals We can investigate the convergence of some series by comparison with an improper integral Consider

7. The Integral Test Suppose c ≥ 0 and f(x) decreasing, positive function with a n = f(n) for all n

8. p-series p-series have the following form Based on what we’ve learned about improper integrals, when will this series converge? –When p > 1 When will it diverge? –When p ≤ 1 We can use this information to show when other series converge or diverge using the comparison test

9. Examples