If you take the sum of "0" from n=1 to infinity, is the series considered converging or diverging?

Question

For example, in a simple geometric series (sum from n=0 to infinity of a*r^n), what if "r" = 0? The sum would definitely equal 0, so according to my knowledge of series, this series would "converge" to 0. Is this correct?

Answer 1

The geometric series as you wrote it converges for 0<|r|<1. You run into a problem with 0⁰ which is undefined.

 

If your sum was n=1 to inf then r=0 would be OK and would converge to 0.

Answer 2

It's a pretty uninteresting series; starts at zero, adds zero over & over, to no effect.  It sure does not diverge.  So let's say it's converging, although we really know that it's like Seinfeld--a series about nothing.