Linked Questions

382
votes
19answers
33k views

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 ...
6
votes
6answers
417 views

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 ...
1
vote
2answers
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.
1
vote
2answers
168 views

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?
-1
votes
3answers
226 views

Evaluate $\sum\limits_{k=1}^{n} \frac{k}{2^k}$ [duplicate]

Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$
0
votes
4answers
77 views

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,...
0
votes
1answer
49 views

How to find the sum of this series? [duplicate]

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....
0
votes
1answer
100 views

Find the value of the series $\sum\limits_{n=1}^ \infty \frac{n}{2^n}$ [duplicate]

Find the value of the series $\sum\limits_{n=1}^ \infty \dfrac{n}{2^n}$ The series on expanding is coming as $\dfrac{1}{2}+\dfrac{2}{2^2}+..$ I tried using the form of $(1+x)^n=1+nx+\dfrac{n(n-1)}{2}...
1
vote
1answer
69 views

How do I find the sum of $\sum\limits_{k=1}^\infty{\frac{k}{2^{k+1}}}=1$? [duplicate]

As shown in the title, how do I find the sum of: $$\sum\limits_{k=1}^\infty{\frac{k}{2^{k+1}}}=1$$
0
votes
1answer
53 views

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

the sum of the series $\sum_{n=1}^\infty \frac{1}{2^{n}}$ is 1. It is easy to find since it is a g.p. the series $\sum_{n=1}^\infty \frac{n}{2^{n}}$ is convergent by ratio test. How will find the ...
2
votes
0answers
60 views

Find $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\frac{2k-1}{2^k}$ [duplicate]

What is the method finding the closed form of $\displaystyle\sum_{k=1}^{n}\frac{2k-1}{2^k}$?
-2
votes
1answer
41 views

How do I find the summation of this series? [duplicate]

$$\sum_{x=0}^{\infty}\frac{x}{2^{x}}$$ This looks like a geometric series but the $x$ is throwing me off. I don't know how to proceed. Any help would be greatly appreciated.
4
votes
6answers
686 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
4
votes
5answers
203 views

How to compute this finite sum $\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$?

I do not know how to find the value of this sum: $$\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$$ (Yes, the last term is added twice). Of course I've already plugged it to wolfram online, and the ...
1
vote
1answer
8k views

Find the value of sum (n/2^n) [duplicate]

I have the series $\sum_{n=0}^\infty \frac{n}{2^n}$. I must show that it converges to 2. I was given a hint to take the derivative of $\sum_{n=0}^\infty x^n$ and multiply by $x$ , which gives $\...

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