MATH128 Study Guide - Final Guide: Ibm System P, Alternating Series, Conditional Convergence

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Calculus ii : practice nal (calculators and textbooks are not allowed during the exam. : evaluate the integral (cid:90) 3w 1 w + 2 (cid:90) 3w 1 (cid:90) (cid:18) w + 2 dw. (cid:19) dw = = 3w 7 ln|w + 2| + c w + 2: find the value of p for which the series is convergent. If p 0, then np = so by the divergent test, this series is divergent for p 0. lim n ln p. If p = 1, then for any positive integer n. since(cid:80) n=1 divergent. For p > 0 and p (cid:54)= 1, let for x 1. 1 n is divergent, by the comparison test, this series is also f (x) = ln x xp: f (x) is positive for x > 1 and continuous on [1, , since f(cid:48)(x) = 1 xp+1 (1 p ln x) so f(cid:48)(x) < 0 for x > e1/p.