Review of Tests for Series

1. Review of Tests for Series

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

3. Since converges, the series converges. Does converge? Justify. When n is large, the function behaves like: positive and finite p- series

4. The Integral Test If is a positive sequence and where is a continuous, positive decreasing function, then: and both converge or both diverge.

5. Does converge? Justify. First show that it is decreasing.

6. 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. NOTE: The nth term test does NOT prove convergence, it only proves divergence!

7. This converges to if , and diverges if . is the interval of convergence. Geometric Series: In a geometric series, each term is found by multiplying the preceding term by the same number, r.

8. Does the series converge or diverge? Find the interval of convergence?.

9. converges if , diverges if . p-series Test

10. Does the series converge or diverge? converge converge diverge diverge converge diverge

11. Direct Comparison Test This series converges. For non-negative series: So this series must also converge. 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 diverge. If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. This series diverges.

12. Does the series converge or diverge? If not for the + 2, this would be a geometric series with r = 4/3 which diverges. Since the + 2 is in the numerator, this series is larger than .

13. For , if then: if the series converges. if the series diverges. if the series may or may not converge. The Ratio Test

14. Does the series converge or diverge?

15. What is the interval of convergence?

16. The interval of convergence is (2,8). The radius of convergence is . The series converges when

17. Alternating Series Test If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series The signs of the terms alternate. Good news! This series converges (by the Alternating Series Test.)

18. This series diverges(by the Alternating Series Test.)

19. converge converge diverge converge converge

20. Assignment Page 506 #1-15 odd

21. Alternating Series Test If the absolute values of the terms approach zero, then an alternating series will always converge! Alternating Series The signs of the terms alternate. Good news! example: This series converges (by the Alternating Series Test.) This series is convergent, but not absolutely convergent. Therefore we say that it is conditionally convergent.

22. If converges, then we say converges absolutely. If converges, then converges. Absolute Convergence The term “converges absolutely” means that the series formed by taking the absolute value of each term converges. Sometimes in the English language we use the word “absolutely” to mean “really” or “actually”. This is not the case here! If the series formed by taking the absolute value of each term converges, then the original series must also converge. “If a series converges absolutely, then it converges.”

23. Alternating Series Estimation Theorem For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term. Since each term of a convergent alternating series moves the partial sum a little closer to the limit: This is a good tool to remember, because it is easier than the LaGrange Error Bound.