HMMT 十一月 2015 · 冲刺赛 · 第 29 题
HMMT November 2015 — Guts Round — Problem 29
题目详情
- [ 15 ] Find the largest real number k such that there exists a sequence of positive reals { a } for which i √ ∑ ∑ ∞ ∞ a n a converges but does not. n k n =1 n =1 n
解析
- [ 15 ] Find the largest real number k such that there exists a sequence of positive reals { a } for which i √ ∑ ∑ a ∞ ∞ n a converges but does not. n k n =1 n =1 n Proposed by: Alexander Katz 1 Answer: 2 1 For k > , I claim that the second sequence must converge. The proof is as follows: by the Cauchy- 2 Schwarz inequality, 2 √ ∑ ∑ ∑ a 1 n ≤ a n k 2 k n n n ≥ 1 n ≥ 1 n ≥ 1 ∑ 1 1 Since for k > , converges, the right hand side converges. Therefore, the left hand side must 2 k n ≥ 1 2 n also converge. 1 1 For k ≤ , the following construction surprisingly works: a = . It can be easily verified that 2 n 2 n log n ∑ a converges, while n n ≥ 1 √ ∑ ∑ a 1 n = 1 2 n log n n n ≥ 1 n ≥ 1 does not converge.