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Possible Duplicate:
How to find the sum of this infinite series

Hello all, I have one last major question, where would I get started on the following question:

$\sum_{n=0}^{\infty}\frac{n}{3^n}$

I know it is a series (obviously), and I think it is geometric, but I have no idea as to how to start it. Does anyone have any first steps/tips as to what I could do for this?

Thanks so much in advance!

Edit: Per the first comment on my posting, by 1hf, see:

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

In particular, see the answer at How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

Thanks all!

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    Thanks! If you put that in a answer form, I can give you credit for the answer easier :)2011-05-02

3 Answers 3

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Hint: Observe that (\sum_{n=0}^\infty x^n)'=\sum_{n=0}^\infty nx^{n-1} and $\sum_{n=0}^\infty x^n$ is convergent for all $|x|<1$

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Let $S_n=\sum_{k=0}^n k/3^k$. Simplify $3 S_{n+1} - S_n$ to determine $S_n$ and then take the limit as $n \to \infty$.

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    I get $3S_{n+1}−S_n = \sum_{k=0}^n 1/3^k$2011-05-02
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It's (not quite) geometric. A geometric series is of the form $\sum_{n = 0}^\infty x^n$.

You are on the right track. For a geometric series, provided $|x|<1$, $\sum_{n = 0}^\infty x^n = \frac{1}{1 - x}.$

For convergent series, it's acceptable to differentiate term by term. This tells us, provided $|x| < 1$, \sum_{n = 0}^\infty nx^{n-1} = \left(\frac{1}{1-x}\right)' = \frac{1}{(1-x)^2}.

I claim your series is very close, but not quite, equal to this form with $x = 1/3$.