27 questions linked to/from Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$?
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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 ...
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### How do you prove $\sum \frac {n}{2^n} = 2$? [duplicate]

How do you prove $$\sum_{n=1}^{\infty} \frac {n}{2^n} = 2\ ?$$ My attempt: I have been trying to find geometric series that converge to 2 which can bind the given series on either side. But I am ...
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### 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.
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### Sum $\sum_{x=0}^{\infty} \frac{x}{2^x}$ [duplicate]

Calculate $\sum\limits_{x=0}^{\infty} \dfrac{x}{2^x}$ So, this series converges by ratio test. How do I find the sum? Any hints?
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### Evaluate $\sum\limits_{k=1}^{n} \frac{k}{2^k}$ [duplicate]

Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$
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### f(x)=(1/2)^x, x=1,2,3,4, …find the mean [duplicate]

A fair coin is flipped successively at random until the first head is observed. Let the random variable X denote the number of flips of the coin that are required. Then the space of x is S={x: x=1,2,...
The series is $1\cdot\frac{1}{2} + 2\cdot\frac{1}{4} + 3\cdot\frac{1}{8} + \cdots$ Or in other words $$\sum_{n=1}^{\infty}\frac{n}{2^n}$$ What kind of series is this and how to find the sum? Thanks....