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Sum of geometric series

  • Thread starter mreaume
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  • #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


[/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!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

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?
 
  • #3
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,508
730

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.
 

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