Calculus 141, section 9.7 Alternating Series, Absolute ...

1 Calculus 141, section Alternating Series, Absolute Convergence notes by Tim Pilachowski So far, we have pretty much limited our attention to series which are positive. What can we say of those which . are not positive? We have taken a quick look at one Alternating series: ( 1)n diverges (Example C in lecture n =1. notes ) because the sequence of partial sums does not converge to a single value, but rather alternates between 1 and 0. The first question becomes: Can we determine whether an Alternating series is convergent or divergent? Theorem , credited to Leibniz, provides a straightforward test. To show that the Alternating . series ( 1)n an or ( 1)n+1 an converges, one need only to show that n =1 n =1.. the sequence {a n } is positive and decreasing, and that lim a n = 0 . n =1 n . Because the terms alternate in sign, the partial sums successively rise and . fall.

2 Because the sequence {a n } is positive and decreasing, the n =1. sequence of partial sums will alternately overshoot and undershoot the limiting value, in a manner similar to the function pictured to the right.. ( 1)n +1 n = 1 2 + 3 K converge? 1 1 1. Example A: Does the Alternating harmonic series Answer: yes n =1.. ( 1)n +1. 1 1 1. Example B: Does the Alternating series = 1 + K converge? Answer: yes 2 4 9. n =1 n . 2n + 1. ( 1)n 3n 1 = 2 + 1 8 + 11 K converge? 3 7 9. Example C: Does the Alternating series Answer: no n =1.. ( 1)n +1 2n 1 = 1 . n 2 3. Example D: Does the Alternating series + K converge? Answer: yes n =1. 3 5. n 2 3 4. Example E: Does the Alternating series ( 1 )n n 1 .. 1 2 3 . = 0 + + K converge? Answer: no n =1 n 2 3 4 . Theorem provides a means of estimating the error involved in using a partial sum to approximate a series: the jth truncation error satisfies E j < a j +1.

3 The proof relies on the Alternating nature of the series. Since the sequence of terms is decreasing, and the partial sums alternately overshoot and undershoot the limit, the error continually decreases by no more than the amount of the succeeding term.. ( 1)n+1 n 1. Example A extended: If we use the Alternating harmonic series to approximate ln 2, which partial n =1. sum would we need to use to be within 10 4 ? Answer: 10,000th partial sum . ( 1)n+1 n 2. 1. Example B extended: Which partial sum would we need to approximate to within 10 4 ? n =1. th Answer: 100 partial sum . For a convergent series a n , if a n converges, then we say that an converges absolutely. n =1 n =1 n =1. Otherwise, it converges conditionally.. ( 1)n +1 n 1. Example A again: Does converge absolutely? Answer: no, conditionally n =1.. ( 1)n +1. 1. Example B again: Does converge absolutely? Answer: yes n =1 n2.

4 ( 1)n +1 = 1 1 1. We can generalize about an Alternating p-series + K , which will converge n =1 np 2p 3p absolutely for all values of p > 1 and diverge for 0 < p 1 . Absolute convergence of a series carries with it a benefit useful in evaluating a series which is neither positive . nor Alternating . By Theorem , if a n converges, then an converges. n =1 n =1.. sin n Example F: Does 2. converge? Answer: yes n =1 n Theorem provides generalized versions of the Direct Comparison Test, Limit Comparison Test, Ratio Test a and Root Test. Corollary states that if either lim n +1 = r < 1 or lim n a n = r < 1 then n an n . lim a n = 0 . Likewise, if r > 1, then lim a n = . n n . The text provides a summary of convergence tests for series at the end of chapter 9, just before the review pages. I have also put a link on the Math 141 webpage to a similar summary which also includes strategies for choosing which test to use in particular circumstances.

5 Example G: Find values for x for which n xn converges. Answer: x < 1. n =1.