I am trying to work out what this series evaluates to:
$$\sum\_{i=1}^\infty i(1-k){k^{i-1}} $$
where k is a constant such that 0 < k < 1.

To figure this out I expand the brackets to get:

$$\sum\_{i=1}^\infty (i{k^{i-1}} -ik^i)$$
which is equivalent to:
$$ \lim\_{N\to \infty}(\sum\_{i=1}^N(i{k^{i-1}} -ik^i)) $$

I then try and write out each term trying to get them to cancel out but I get:

$$ \lim\_{N\to \infty} (1 -k+2k-2k^2+3k^2-...+Nk^{N-1}-(N-1)k^{N-1}-Nk^N)$$ = $$ \lim\_{N\to \infty}(\sum\_{i=0}^{N-1} k^i)+\lim\_{N\to \infty}(-Nk^N)$$

I think I want to use the fact that $$\left\lvert k \right\rvert<0$$ to show that the larger terms go to zero but I am unclear how to do this. Thanks for any help in advance.