## On Convergent Sequences

So here's a question that I've been preoccupied with in the last couple of days. Suppose I have a convergent (at the sum) sequence, if you like a geometric sequence, say . If one were to multiply the elements of the sequence by its corresponding index, would this still be a convergent sequence? In this case, I'm concerned with and its convergence (at the sum). I wish to explore whether there are particular circumstances in which this is indeed the case.

I'll leave the question open ended for a bit. I have solved this via derivative considerations, I think (I do not mean it as a pun... I think I have solved this by considering derivatives of functions).

Categories: Combinatorics and Probability, Infinite Sums, Mathematics