Calc limit of a series My HW is
$$\lim\limits_{n\to\infty}\dfrac{1\cdot x+2\cdot x^2+3\cdot x^3+\dots+n\cdot x^n}{n\cdot x^{n+1}}$$ as $$x\gt1$$
I've multiplied both numerator and denominator by $$1/x^n$$
so my answer $$1/x$$ 
I was told that it is not correct.
Can anybody please help? 

Rewrite the numerator as
$$
\sum_{i=1}^n \sum_{j=0}^{n-i} x^j = \sum_{i=1}^n \frac{x^{n-i}-1}{x-1}=\frac{1}{x-1}\left(\frac{x^n-1}{x-1}-\frac{n(n+1)}{2}\right).
$$
It follows that the limit is equal to
$$
\lim_{n\to \infty}\frac{1}{nx^{n+1}}\left(\frac{x^n-1}{(x-1)^2}-\frac{n(n+1)}{2(x-1)}\right).
$$
Both summands tend to $0$.
