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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 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!


3 Answers 3


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 $|x|<1$


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 $n \to \infty$.

  • $\begingroup$ 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? $\endgroup$
    – quanta
    Commented May 2, 2011 at 16:59
  • 1
    $\begingroup$ I get $3S_{n+1}−S_n = \sum_{k=0}^n 1/3^k$ $\endgroup$
    – Emre
    Commented May 2, 2011 at 17:10

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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