- #1

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## Homework Statement

Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf).

## Homework Equations

I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1.

## The Attempt at a Solution

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I began by rescaling the sum, i.e.

Σ (n+1)*(1/2)^(n+1) (from n = 0 to n = inf)

= Σ (n+1)*(1/2)*(1/2)^n (from n = 0 to n = inf)

= (1/2) ∑ (n+1)*(1/2)^n (from n = 0 to n = inf)

= (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + ∑ (1/2)^n (from n = 0 to n = inf) )

= (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + 1 / (1 - 1/2) )

= (1/2) ( ∑ n*(1/2)^n (from n = 0 to n = inf) + 2 )

I'm stuck here. I don't know how to evaluate the first summation. Any help would be appreciated!