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\(\frac{1}{2^1} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots + \frac{k}{2^k} + \cdots \)

 Nov 4, 2020
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Let this sum be $S.$

 

$S = 1/2 + 2/4 + 3/8 + n/2^n$

Multiplying by two:

$2S = 1 + 1 + 3/4 + ... 2n/2^n$

 

We subtract these, we can shift the infinite series over one term:

 

$S = 1 + 1/2 + 1/4 + ... + 1/2^n$

 

Using the geometric series formula, $\frac{1}{1-1/2} = \boxed{2}$

 Nov 4, 2020

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