MATH 1B Lecture Notes - Lecture 17: Ratio Test
Document Summary
Math 1b: calculus - lecture 17: introduction to power series. A: consider the sum from n=0 to of xn(n! Then the limit as n of |(an+1) / an | = lim n |[(n+1)! xn+1] / [(n!) xn] = lim n |(n+1)x|. So, lim n |an+1 / an| = provided x 0. Hence, by the ratio test, the sum of n=0 to of xn(n!) diverges when x 0. If x=0, then the sum of n=0 to of xn(n!) converges. A: consider the sum fro n=0 to of xn. This is a geometric series with first term 1 and common ratio x. So, the series converges when |x| < 1, and diverges otherwise. A: consider the sum from n=0 to of (xn / n! Then the limit as n | (xn+1 / (n+1)!) / xn) | = lim n |x /( n+1)| = 0, which is < 1.
