Find a constant k such that for any positive a_{i}, ∑_{1}^{∞} n/(a_{1} + a_{2} + ... + a_{n}) ≤ k ∑_{1}^{∞}1/a_{n}.

**Solution**

Let us write s_{n} = a_{1} + a_{2} + ... + a_{n} and b_{n}= n/s_{n}. If ∑ 1/a_{n} diverges, then there is nothing to prove. So assume it converges. Hence a_{n} must diverge, so there are only finitely many values less than any given K, so we can arrange the terms in increasing order c_{1} ≤ c_{2} ≤ c_{3} ... . Since all the terms are positive ∑ 1/a_{n} is absolutely convergent and equal to ∑ 1/c_{n}. Also we have s_{n} ≥ c_{1} + c_{2} + ... + c_{n}. The trick is to notice that half the c_{i} are at least c_{n/2}. It is convenient to consider separately n = 2m and n = 2m+1. So s_{2m} ≥ m c_{m}. Hence b_{2m} < 2/c_{m}. Similarly, s_{2m-1} > m c_{m}, so b_{2m-1} < 2/c_{m}. Hence ∑_{1}^{2m} b_{i} ≤ 4 ∑_{1}^{m} 1/c_{i}, which gives the required result with k = 4.

© John Scholes

jscholes@kalva.demon.co.uk

5 Feb 2002