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I have some examples for this problem in my notes, but do not have the whole method shown step by step, and can not figure it out.

Take as an example:$$ \sum_{n=1}^{\infty} n 3^{-n} $$

I only have in my notes written the result, which is that this series converges, and the sum is .75; but can't figure out how to calculate the formula for partial sums, which I then calculate the limit of.

Matti P.
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Bozont
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    Hint: $$ \sum\limits_{n = 1}^N {nx^n } = x\frac{d}{{dx}}\left( {\sum\limits_{n = 1}^N {x^n } } \right) = \cdots $$ – Gary Jan 19 '21 at 08:54
  • I agree with Gary. If you know how to calculate $\sum 3^{-n}$, then do that first and then essentially take the derivative. That is indeed the trick here. – Matti P. Jan 19 '21 at 08:56

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