9.4 Part 1 Convergence of a Series

1. 9.4 Part 1 Convergence of a Series

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

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

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

5. Direct Comparison Test For non-negative series: This series converges. If every term of a series is less than the corresponding term of a convergent series, then both series converge. So this series must also 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 Test For non-negative series: So this series must also diverge. If every term of a series is greater than the corresponding term of a divergent series, then both series 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. The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge. Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve: This leads to:

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

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

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

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

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

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

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