>> One of my students asked me today a question that I was unable to >> answer. Let _f_ be an analytical function from (0,+oo) into [1,+oo) and >> suppose that the integral of _f_ from 1 to +oo converges. Does it follow >> that the series sum_n f(n) converges? I don't think so, but I was unable >> to find a counter-example. Any ideas? > > One can imagine an analytic function which is equal to 1 at every > natural number but such that the sequence of its integrals from n-1/2 to > n+1/2 converges. > > I do not have a concrete example in mind but I'm certain that it is > possible. > > It could easily be derived from an analytic function with value 0 > outside [-0.5 , .5] and value 1 at 0.