AP Calculus Miss Battaglia

 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:

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

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, S n, 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.

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.

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

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

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

Find the sum of the series

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