HMMT 十一月 2015 · 冲刺赛 · 第 29 题

英文解析

29. [ 15 ] Find the largest real number k such that there exists a sequence of positive reals { a } for whichi
    √
    ∑ ∑
    ∞ ∞ana converges but does not.
    n kn =1 n =1
    Proposed by: Alexander Katzn
    Answer:1
    For k > , I claim that the second sequence must converge. The proof is as follows: by the Cauchy-12
    Schwarz inequality,2
         
    √2
    ∑ ∑ ∑
    a 1
         n
    ≤ ank 2 kn nn ≥ 1 n ≥ 1 n ≥ 1
    ∑
    1 1
    Since for k > , converges, the right hand side converges. Therefore, the left hand side must
    2 kn ≥ 1
    2 nalso converge.
    1 1
    For k ≤ , the following construction surprisingly works: a = . It can be easily verified thatn 2
    2 n log n
    ∑
    a converges, whilenn ≥ 1
    √
    ∑ ∑
    a 1 =n
    2 n log n 1
    n ≥ 1 n ≥ 1 ndoes not converge.