is a divergent series of positive terms, let 
for n=1,2,…. For which values of p does the series 
converge?Solution OF Elementary Problem
E 2558. Proposed By a. Torchinsky, Cornell University
Suppose that is a divergent series of positive terms, let for n=1,2,…. For which values of p does the series converge?
Solution by Elmer K. Hayashi. We prove a more general theorem from which we deduce that if and only if p > 1.
Theorem. Let f(x), for x > 0, be any nonnegative, continuous, monotonically decreasing, real-valued function. if is a divergent series of positive terms and if for n=1,2,… , then
converges if
,
and
diverges if 
.
Proof:………