TITLE: Sum of n*constant^n from 1 to N
QUESTION [0 upvotes]: What is the value of the sum for n=1 to n=N of n * constant ^ n?
I have considered Geometric but with the multiplication by n I cannot get this to work.

REPLY [0 votes]: We have that 
$$\sum_{n=1}^N{nx^n}=x\sum_{n=1}^N{nx^{n-1}}=x\sum_{n=1}^N{\frac{d}{dx}x^n}=x\cdot\frac{d}{dx}\left[\sum_{n=1}^N{x^{n}}\right]$$
There is your geometric series.  Multiply your result of the series by $x$ and then differentiate with respect to $x$.
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