MATH101 Study Guide - Final Guide: Ratio Test, The Sketch, Absolute Convergence

• School: University of Alberta
• Department: Mathematics
• Course: MATH101
• Professor: All

Document Summary

Solutions for math 101 2006 final: part (a). 1 x2 (1 + x2) dx =z 1 x2 (cid:0) 1 + x2 dx = (cid:0)x(cid:0)1 (cid:0) tan(cid:0)1 (x) + c: 1 x2 (1 + x2) dx = lim t!1z t. 4 (cid:0) lim t!1(cid:2)x(cid:0)1 + tan(cid:0)1 (x)(cid:3)t t!1(cid:2)t(cid:0)1 + tan(cid:0)1 (t)(cid:3) = 1 (cid:0) 1 x2 (1 + x2) dx = lim t!0+z 1 t. = (cid:0) lim t!0+(cid:2)x(cid:0)1 + tan(cid:0)1 (x)(cid:3)1 t!0+(cid:2)t(cid:0)1 + tan(cid:0)1 (t)(cid:3) = 1; =) x ln (1 (cid:0) 2x) = (cid:0) We apply the ratio test to the series. 1 yielding lim n!1(cid:12)(cid:12)(cid:12)(cid:12) an+1 n!1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) an (cid:12)(cid:12)(cid:12)(cid:12) = lim. = lim n!1(cid:12)(cid:12)(cid:12)(cid:12) (n + 1) 2x n + 2 (cid:12)(cid:12)(cid:12)(cid:12) = j2xj : For absolute convergence we must have j2xj < 1 () jxj < 1=2 =) r = 1=2: For the full interval of convergence we must test the end-points given by x = (cid:6)1=2. 1 n which is divergent by the p-test.