Sum of a sequence

**lausing** · February 21st 2010, 09:45 PM

$\sum_{x=0}^{\infty}\frac{x}{2^x}$

All I know is that it converges. Really have nothing more to say.

**Soroban** · February 21st 2010, 10:05 PM

Hello, lausing!

$S \;=\;\sum^{\infty}_{x=1} \frac{x}{2^x}$

$\begin{array}{cccccc}\text{We have:} & S &=& \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \hdots \\ \\[-3mm]<br /> \text{Multiply by }\frac{1}{2}\!: & \frac{1}{2}S &=& \quad\;\;\dfrac{1}{2^2} + \dfrac{2}{2^3} + \dfrac{3}{2^4} + \hdots \end{array}$

. . $\text{Subtract: }\;\;\tfrac{1}{2}S \;\;=\;\;\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots$ .[1]

The right side is a geometric series with: . $a = \tfrac{1}{2},\;\;r = \tfrac{1}{2}$

. . Its sum is: . $\frac{\frac{1}{2}}{1-\frac{1}{2}} \:=\:1$

Hence [1] becomes: . $\tfrac{1}{2}S \;=\;1$

Therefore: . $S \;=\;2$

**lausing** · February 21st 2010, 10:14 PM

Thanks!

Also, the question is erroneously posted in calculus rather than linear algebra or something, since it orginally was about derivatives. Whoops!

Thread: Sum of a sequence

LinkBack

Thread Tools

Display

Sum of a sequence

Similar Math Help Forum Discussions

Arithmetic Sequence problem and sumac sequence

Randomized sequence of blocked 1-0-sequence and assignment of random integers

Recursive sequence (Fibonacci Sequence) & Limits

sequence membership and sequence builder operators

set, geometric sequence,fraction and arithmetic sequence

Search Tags