MAT 21C Lecture Notes - Lecture 5: Integral Test For Convergence, Ibm System P, Telescoping Series

• School: UC-Davis
• Department: Mathematics
• Course: MAT 21C
• Professor: John Hunter

Document Summary

Mat 21c lecture 5 integral test and comparison tests. The integral test implies that the p-series converges if p > 1 and diverges if 0 < p. < 1: two important types of series, problem: apply the integral test to prove for the p-series, the geometric series grow more rapidly than the p-series to converge. Integral test suppose that (cid:1853)=(cid:4666)(cid:4667) where f(x) > 0 is a positive, decreasing converges if (cid:4666)(cid:4667)(cid:1856) continuous function for x > 1. =(cid:2869) diverges if (cid:4666)(cid:4667)(cid:1856) converges and b) . = 1 + (cid:2869)(cid:2870) + (cid:2869)(cid:2871) + (cid:2869)(cid:2872) + (cid:1853)= (cid:2869) = f(n); f(x) = (cid:2869) (cid:2869) Applying the integral test, we have (cid:2869)(cid:1856) (cid:2869) lim(cid:3029) (cid:3029)(cid:3117) (cid:2869) (cid:2869)(cid:2869) for p 1 so (cid:2869)(cid:1856) < if p > 1 and (cid:2869)(cid:1856) = if 0 < p < 1. (cid:2869: 1) geometric series: converges if || < 1 and diverges if || > 1. = (cid:2869)(cid:2870) + (cid:2869)(cid:2872) + (cid:2869)(cid:2876) + (cid:2869)(cid:2869)(cid:2874)+ .