use the geometric series + L'Hospital's rule
$$ \begin{align} \sum_{i=0}^{n} x^{i} &= \frac{x^{n+1}-1}{x-1} \ \frac{d}{dx}(\sum_{i=0}^{n} x^{i}) &= \frac{d}{dx}(\frac{x^{n+1}-1}{x-1}) \ \sum_{i=1}^{n} ix^{i-1} &= \frac{nx^{n+1}-(n+1)x^n+1}{(x-1)^2} \end{align}