Sum of Geometric Series by Differentiation

[mreaume]

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

[/B]
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!

 

[Dick]

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

[/B]
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!

Try taking the derivative with respect to r of both sides of Σ r^n (from n = 0 to n = inf) = 1 / (1 - r). Can you relate that result to your problem?

 

[LCKurtz]

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

[/B]
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!

Hint: You know $s = \sum r^n$. Think about differentiating that with respect to $r$ and working with that.