1. MTH253 Calculus III Chapter 11, Part I (sections 11.1 – 11.6) Sequences Series Convergence Tests

2. Sequences Formal ◦ A function whose domain is a set of integers. Intuitive ◦ An ordered list of numbers (terms). General Term Useful Tips (assume n starts at 1) : integers starting at k = n+(k-1) evens = 2n odds = 2n-1 alternating signs (1 st positive): (-1) n+1 alternating signs (1 st negative): (-1) n

3. Sequences - Examples

4. Limit of a Sequence Converges if Diverges if

5. Tests for Increasing/Decreasing Difference Between TermsRatio of Successive Terms Derivative

6. Convergence of Sequences Monotonic & Bounded Examples Eventually decreasing and always positive. (why?) Therefore it converges. (why?) Eventually increasing but no upper bound. (why?) Therefore it diverges. (why?)

7. Series The sum of the terms of a sequence. Sequence of Partial Sums

8. Convergence of a Series If the sequence of partial sums converges, then the series converges to that limit. Example: NOTE: A general expression for s n is usually difficult to determine.

9. 10 Convergence Tests n th Term Test (Divergence Test) Geometric Series Test P-Series Test Integral Test Comparison Test Limit Comparison Test Ratio Test Root Test Alternating Series Test Absolute Convergence Test Why so many tests? Because each has its limitations (i.e. none of them work for all series).

10. Proof !? A proof is a step by step argument that includes reasons why each statement in the argument follows from the previous statement or statements. Reasons are needed unless a conclusion is “obvious”. In this chapter, we are proving that a series converges or diverges based on known or previously proven facts.

11. n th Term Test (Divergence Test) If … then the series diverges. Otherwise, the test fails.

12. Example …

13. Geometric Series Test If … then the series converges if … Otherwise, the series diverges. NOTE: This test fails when the series is not a geometric series.

14. Example …

15. P-Series Test If … then the series converges if … Otherwise, the series diverges. NOTE: This test fails when the series is not a P-Series. If p = 1, the series diverges. This is a special series called the “Harmonic Series”

16. Example …

17. Integral Test For any function evaluate The integral and the series will have the same behavior (i.e. they either both converge or both diverge). NOTE: This test fails when the integral cannot be determined.

18. Example …

19. Comparison Test NOTE: This test fails when an appropriate series can’t be found. Ifandconverges, converges.then Ifanddiverges, diverges.then

20. Using the Comparison Test How do you find a series to compare? If the general term is a fraction … ◦ Increasing the numerator makes it bigger. ◦ Decreasing the numerator makes is smaller. ◦ Increasing the denominator makes it smaller. ◦ Decreasing the denominator makes it bigger. So, make several little changes that either increase or decrease the series until you get a series whose behavior you know. ◦ Inc. the numerator & dec. the denominator,,, bugger, ◦ Dec. the numerator & inc. the denominator,,, smaller.

21. Example …

23. Limit Comparison Test NOTE: This test fails when an appropriate series can’t be found. Given a series of known behavior …

24. Using the Limit Comparison Test How do you find a series to compare? Use a similar but simpler series whose behavior you can determine. ◦ If all or a piece of the series is a polynomial, discard all but the leading term. ◦ Discard constant multiples and/or constant terms.

25. Example …

26. Ratio Test If … then … if L < 1, then the series converges if L > 1, then the series diverges if L = 1, then the test fails.

27. Example …

28. Root Test If … then … if L < 1, then the series converges if L > 1, then the series diverges if L = 1, then the test fails.

29. Example …

30. Alternating Series Test For a series of the form … and If … … then the series converges, otherwise it diverges. NOTE: This test fails when the series is not alternating.

31. Example …

32. Absolute Convergence Test If the series of absolute values converges … … then the series converges, otherwise the test fails. NOTE: In this case, the original series is said to be Absolutely Convergent or it Converges Absolutely.

33. Example …