Practice Assignment 5 - First-Year Interest Group Seminar | N 1, Assignments of Health sciences

Download Practice Assignment 5 - First-Year Interest Group Seminar | N 1 and more Assignments Health sciences in PDF only on Docsity! MATH 361K – HOMEWORK ASSIGNMENT 5 Due Thursday, March 5, 2009 Please write clearly, and staple your work ! 1. Problem If ∑ xn with xn > 0 is convergent, then is ∑ x2n always convergent ? Either prove it, or give a counterexample. 2. Problems Assume that the series ∑ xn, with xn > 0, is convergent. Defining yn := 1 n(x1 + · · ·+ xn), prove that ∑ yn is always divergent. 3. Problems Assume ∑∞ n=1 xn where (xn) with xn > 0 is a strictly decreasing sequence of real numbers. let sn := ∑n k=1 xn denote the n-th partial sum. Prove that 1 2 (x1 +2x2 +4x4 + · · ·+2nx2n) ≤ s2n ≤ (x1 +2x2 + · · ·+2n−1x2n−1)+x2n . Use these inequalities to prove that ∑∞ n=1 xn converges if and only if ∑ 2nx2n converges. This is the so-called Cauchy condensation test. 4. Problem Use the Cauchy condensation test to prove that the p-series ∑ 1 np con- verges for all p > 1. 5. Problem Use the Cauchy condensation test to prove that ∑ 1 n ln n diverges. 1