MATH 231 Lecture Notes - Lecture 18: Absolute Convergence, Limit Comparison Test, Alternating Series

• School: University of Illinois
• Department: Mathematics
• Course: MATH 231
• Professor: Bob Muncaster

Document Summary

Power series f (x)=c 0+c1 x+c2 x2+c 3 x3+ Previously, we have called this an infinite series. It is actually a power series centered at x. General form of a power series x a 3+ x a 2+c 3 g(x)=c0+c (x a)+c 2 x a n. A power series is centered at x=a. C n could possibly a. ) converge only for x = a. By the ratio test lim n an +1 an = n (n+1)! xn +1 n! xn lim. We find that the series will only converge absolutely if the limit of the series equals 0, and this can only happen when x=0. This holds true since we already identified a=0. b. By the ratio test lim n an +1 an = n n! xn+1 (n+1)! xn =lim lim n . The limit of the series will always equal zero, regardless of the value of x.