Sum of a sequence

• Feb 21st 2010, 10:45 PM
lausing
Sum of a sequence
$\sum_{x=0}^{\infty}\frac{x}{2^x}$

All I know is that it converges. Really have nothing more to say.
• Feb 21st 2010, 11:05 PM
Soroban
Hello, lausing!

Quote:

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

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

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

• Feb 21st 2010, 11:14 PM
lausing
Thanks!

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