How can I evaluate \sum_{n=0}^\infty (n+1)x^n

How can I evaluate

n=12n3n+1
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before so I feel that there must be a simpler method.
In general, how can I evaluate
n=0(n+1)xn?

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