Linked Questions

452 votes
24 answers
90k 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 ...
backus's user avatar
  • 4,735
0 votes
2 answers
225 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?
6c656c's user avatar
  • 283
1 vote
2 answers
1k 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.
user10024395's user avatar
1 vote
1 answer
678 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 (...
Bernd's user avatar
  • 809
0 votes
4 answers
97 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 ...
Kurt Peek's user avatar
  • 197
10 votes
1 answer
44k 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 $\...
Dr. John A Zoidberg's user avatar
1 vote
2 answers
5k 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?
square_one's user avatar
  • 2,357
6 votes
3 answers
4k views

Sum of $\sum_{n=1}^{\infty}\frac{n}{2^n}$. [duplicate]

I was trying to find the sum of $\sum_{n=1}^{\infty}\frac{n}{2^n}$. I tried like this $S = \sum_{n=1}^{\infty}\frac{n}{2^n} = \sum_{n=1}^{\infty}(\frac{n+1}{2^n} - \frac{1}{2^n}) = 2\sum_{n=2}^{\...
BAYMAX's user avatar
  • 4,992
2 votes
4 answers
1k views

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?
bsky's user avatar
  • 500
3 votes
3 answers
477 views

How to prove $\sum\limits_{i=1}^{\infty}\frac{i}{2^i}$ converges? [closed]

What would be the simplest way to prove that $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges?
crgolden's user avatar
  • 141
2 votes
3 answers
528 views

Show that $\sum_{n=0}^{\infty} \frac{n}{2^{n+1}} = 1$

My Work I felt the best way to go about this problem was to compare it to a well known MacLaurin series. I noticed it resembled the reciprocal of the absolute value of the MacLaurin series of $\ln(1+...
Dunka's user avatar
  • 2,787
3 votes
2 answers
297 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.
Iching Sebastian Quares's user avatar
-1 votes
3 answers
403 views

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

Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$
sasza90's user avatar
  • 19
1 vote
3 answers
155 views

Evaluate to find the sum of an infinite series [duplicate]

$∑_{n=1}^\infty$ $n\over2^{n-1}$ or 1 + $2\over2$ + $3\over4$ + $4\over8$ + $5\over16$ + $\ldots$ How to go about evaluating the above, showing that it sums to 4?
Jack's user avatar
  • 271