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### How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?

How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
171 views

### Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$ [duplicate]

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?
574 views

### Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum $\sum_{i=0}^{\infty} \frac{i}{2^i}$ which I know should be equal to $2$, but I cannot prove it. If I am not mistaken, it should be a arithmetico-geometric series (...
473 views

### How to evaluate the following series [duplicate]

Determine the sum of $$\sum_n^\infty \frac{k}{3^k}$$ Can someone teach me how to solve this please thanks.
77 views

### How to compute this infinite sum? [duplicate]

I'm trying to compute the infinite sum $\sum_{n=1}^{\infty}n(\frac{1}{2})^n$ which I believe should represent the expected amount of coin flips needed to get a head. Can someone remind me how to do ...
8k views

139 views

### Sum of the series $\sum \frac{n}{2^{n}}$ [duplicate]

I know that the series converges by d'Alembert ratio test, where $\lim\left ( \frac{A_{n+1}}{A_{n}} \right )= \frac{1}{2}$, but I don't know how to calculate the sum of the serie. Thanks for the help.
### Prove limit of $\sum_{n=1}^\infty n/(2^n)$ [duplicate]
How do you prove the following limit? $$\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{k}{2^k}\right)=2$$ Do you need any theorems to prove it?