Let $|r| < 1$,

Question

$$S = \sum_{k=0}^{\infty} r^k,$$
and
$$T = \sum_{k=0}^{\infty} k r^k.$$
Our approach is to write $T$ as a geometric series in terms of $S$ and $r$.

Give a closed form expression for $T$ in terms of $r$.

Answer 1

google is your friend.

A discussion of this topic is found here

https://www.physicsforums.com/threads/sum-of-k-x-k.372124/

and wolframalpha.com says that

T = r/(1-r)^2

I'm sure you can find other interesting explorations online.

Answer 2

this is correct

Answer 3

stop cheating on aops

whoever you are, you searched up this question and you're in my intermediate algebra class and you're doing arithmetic sequences and series

stop this

now

Answer 4

why are you on this then