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On Convergent Sequences

November 13th, 2016 No comments

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  1/2, 1/4, 1/8, ... = 1.  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 1/2 * 1, 1/4 * 2, 1/8 * 3, ... 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).