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2017 PUMaC Individual Finals A2

Source:

September 20, 2019
algebra

Problem Statement

Let a1,a2,a3,...a_1, a_2, a_3, ... be a monotonically decreasing sequence of positive real numbers converging to zero. Suppose that Σi=1∞aii\Sigma_{i=1}^{\infty}\frac{a_i}{i} diverges. Show that Σi=1∞ai22017\Sigma_{i=1}^{\infty}a_i^{2^{2017}} also diverges. You may assume in your proof that Σi=1∞1ip\Sigma_{i=1}^{\infty}\frac{1}{i^p} converges for all real numbers p>1p > 1. (A sum Σi=1∞bi\Sigma_{i=1}^{\infty}b_i of positive real numbers bib_i diverges if for each real number NN there is a positive integer kk such that b1+b2+...+bk>Nb_1+b_2+...+b_k > N.)
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