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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:


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 to find the sum of this infinite series

In particular, see the answer at How to find the sum of this infinite series

Thanks all!

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marked as duplicate by t.b., Arturo Magidin, Aryabhata, Asaf Karagila, Sivaram May 2 '11 at 17:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Very close to… . In particular, see the answer at… – lhf May 2 '11 at 16:41
Thanks! If you put that in a answer form, I can give you credit for the answer easier :) – Nitroware May 2 '11 at 16:46

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 $3 S_{n+1} - S_n = 2 S_n + \frac{n+1}{3^n}$, in the limit this is $2S = 2S$... How do you get the sum? – quanta May 2 '11 at 16:59
I get $3S_{n+1}−S_n = \sum_{k=0}^n 1/3^k$ – Emre May 2 '11 at 17:10
I see! Thanks – quanta May 2 '11 at 18:48

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 $x = 1/3$.

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