9-2 Series

1. 9-2 Series AP Calculus Miss Battaglia

2. Summing Series • An infinite series (or just a series for short) is simply adding up the infinite number of terms of a sequence. Consider: The series associated with the sequence is:

3. You can use fancy summation notation to write this sum in a more compact form:

4. Partial Sums Continuing with the same series, look at how the sum grows by listing the “sum” of one term, two terms, three terms, etc. The nth partial sum, Sn, of an infinite series is the sum of the first n terms of the series. If you list the partial sums, you have a sequence of partial sums.

5. The Main Event: Convergence and Divergence of a Series If the sequence of partial sums converges, you say that the series converges; otherwise, the sequence of partial sums diverges and you say that the series diverges.

6. A no-brainer divergence test: The nth term test

7. Geometric Series A geometric series is a series of the form: The first term, a, is called the leading term. Each term after the first equals the preceding term multiplied by r, which is called the ratio. Ex: a = 5 and r = 3

8. Geometric Series Rule If 0 < |r| < 1, the geometric series converges to . If |r| > 1, the series diverges. Ex: a = 5 and r = 3

9. Convergent and Divergent Geometric Series

10. Telescoping Series To see that this is a telescoping series you have to use partial fractions. A telescoping series will converge iffbn approaches a finite number as n∞. Moreover, if the series converges it sum is

11. Writing a Series in Telescoping Form Find the sum of the series

12. Homework • P. 614 #9, 12, 17, 18, 37, 39, 43, 45, 51, 59, 61, 68, 69, 70, 71, 72, 74