Finding sum of the power series and the sum of the series [duplicate]

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(1) Find the sum of the power series

$$\sum_{n=1}^{\infty} nx^n$$

(2) Find the sum of the series

$$\sum_{n=1}^{\infty} \frac{n}{3^n}$$

Any tips on solving the sum of series/power series?

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2 Answers

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You can find the second sum using the first with $x = 1/3$. As for the first, for $|x| <1$ write $$ \sum_{n=1}^\infty n x^n = x \sum_{n=1}^\infty n x^{n-1} = x \sum_{n=1}^\infty \frac{d}{dx} x^n = x \frac{d}{dx} \sum_{n=1}^\infty x^n = x \frac{d}{dx} \frac{x}{1-x} = \frac{x }{(1-x)^2}. $$

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Hint: If you take the power series $1/(1-x) = \sum_{n=0}^\infty x^n$ and take the derivative term by term, what do you get? Also the second series is a special case of the first where $x = 1/3$.

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