1. 9.4 Part 1 Convergence of a Series

2. The first requirement of convergence is that the terms must approach zero. n th term test for divergence diverges if fails to exist or is not zero. Note that this can prove that a series diverges, but can not prove that a series converges.

3. The first requirement of convergence is that the terms must approach zero. n th term test for divergence diverges if fails to exist or is not zero. So eventually, as n  ∞, the sum goes to 1 + 1 + 1 + 1… So the series diverges

4. So eventually, as n  ∞, the sum goes to 0 + 0 + 0 + 0… The first requirement of convergence is that the terms must approach zero. n th term test for divergence diverges if fails to exist or is not zero. So the series may converge

5. This series converges. So this series must also converge. Direct Comparison TestFor non-negative series: If every term of a series is less than the corresponding term of a convergent series, then both series converge. is a convergent geometric series But what about… for all integers n > 0 So by the Direct Comparison Test, the series converges

6. Direct Comparison TestFor non-negative series: If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. So this series must also diverge. This series diverges. is a divergent geometric series But what about… for all integers n > 0 So by the Direct Comparison Test, the series diverges

7. Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to: The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge.

8. Example 1: Does converge? Since the integral converges, the series must converge. (but not necessarily to 2.)

9. p-series Test converges if, diverges if. We could show this with the integral test. If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

10. the harmonic series: diverges. (It is a p-series with p=1.) It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series. Notice also that the terms go to 0 yet it still diverges

11. Limit Comparison Test If and for all ( N a positive integer) If, then both and converge or both diverge. If, then converges if converges.If, then diverges if diverges.

12. Example : When n is large, the function behaves like: Since diverges, the series diverges. harmonic series

13. Example 3b: When n is large, the function behaves like: Since converges, the series converges. geometric series

14. Another series for which it is easy to find the sum is the telescoping series. Ex. 6: Using partial fractions:

15. Telescoping Series converges to Another series for which it is easy to find the sum is the telescoping series. Ex. 6: 