- #1
mreaume
- 11
- 0
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!