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Sec. 9.2: Series and Convergence

Sec. 9.2: Series and Convergence. Sec. 9.2: Series and Convergence. If { a n } represents a sequence , then the sum of the terms of a sequence represents an infinite series (or simply a series ). Sec. 9.2: Series and Convergence.

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Sec. 9.2: Series and Convergence

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  1. Sec. 9.2: Series and Convergence

  2. Sec. 9.2: Series and Convergence If {an} represents a sequence, then the sum of the terms of a sequence represents an infinite series (or simply a series).

  3. Sec. 9.2: Series and Convergence To find this sum, consider the sequence of partial sums: S1 = a1 S2 = S1 + a2 = a1 + a2 S3 = S2 + a3 = a1 + a2 + a3 . . . Sn = Sn – 1 + an = a1 + a2 + a3 + . . . + an If this sequence of partial sums converges, then the series is said to converge.

  4. Sec. 9.2: Series and Convergence

  5. Sec. 9.2: Series and Convergence When dealing with series, there are two basic questions to consider: (1) Does a series converge or diverge? (2) If a series converges, what is its sum?

  6. Sec. 9.2: Series and Convergence Ex: Determine the convergence / divergence of the following series: List the partial sums: The Series converges and its sum = 1.

  7. Sec. 9.2: Series and Convergence Ex: Determine the convergence / divergence of the following series: The nth partial sum is

  8. Sec. 9.2: Series and Convergence This is an example of a telescoping series. It is of the form Thenth partial sum is Sn = b1 – bn + 1; it follows that a telescoping series will converge if and only if bn approaches a finite number as n→ ∞.

  9. Sec. 9.2: Series and Convergence Ex: Find the sum of the series Method #1: Partial Sums

  10. Sec. 9.2: Series and Convergence Ex: Find the sum of the series Method #2: Partial Fractions

  11. Sec. 9.2: Series and Convergence Ex: Find the sum of the series Method #2: Partial Fractions

  12. Sec. 9.2: Series and Convergence Sum: S = b1–L

  13. Sec. 9.2: Series and Convergence Geometric Series A geometric series is of the form with a common ratio r.

  14. Sec. 9.2: Series and Convergence

  15. Sec. 9.2: Series and Convergence Ex: Determine the convergence / divergence of the following geometric series. If the series converges, find its sum. Identify r and a. State this.

  16. AP Calculus BCMonday, 25 March 2013 • OBJECTIVETSW (1) understand the definition of a convergent infinite series; (2) use properties of infinite geometric series; and (3) use the nth-Term Test for Divergence of an infinite series. • ASSIGNMENT DUE • Sec. 9.1 • If you indicated that you want a calculus chart, bring $6.50 on Monday. When I get all monies, I will combine with Ms. Barron’s students and place the order. • REMINDER • Projects due tomorrow, 26 March 2013 at the beginning of the period.

  17. Sec. 9.2: Series and Convergence Ex: Determine the convergence / divergence of the following geometric series. If it converges, find its sum.

  18. Sec. 9.2: Series and Convergence Ex: Use a geometric series to represent as the ratio of two integers. This can be written as a series: a

  19. Sec. 9.2: Series and Convergence Sum:

  20. Sec. 9.2: Series and Convergence

  21. Sec. 9.2: Series and Convergence

  22. Sec. 9.2: Series and Convergence

  23. Sec. 9.2: Series and Convergence Ex: Apply the nth-Term Test to the following: The limit ≠ 0, so the series diverges. Test: nTT Test: nTT The limit ≠ 0, so the series diverges. Another test must be used. The limit = 0; the nth-Term Test fails.

  24. Sec. 9.2: Series and Convergence This test cannot be used to show convergence

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