Re: how to find out the closed formula for this sequence?

From: Torsten Hennig (Torsten.Hennig_at_umsicht.fhg.de)
Date: 02/02/05 

Date: Wed, 2 Feb 2005 12:48:53 +0000 (UTC)

On 02 Feb 2005, wrote:
>Hello there
>
>I would like to get a closed form for the sequence
>
>1/(1+r) + 2/(1+r)^2 + 3/(1+r)^3 + ... + n/(1+r)^n
>Could anyone out there help me out?
>
>thank you very much!!

Hi,

from the geometric series you have

sum (k=0,...,n) x^k = (1-x^(n+1))/(1-x).

Differentiating this identity with respect to x and multiplying
both sides with x leads to

sum (k=1,...,n) k*x^k = x* d/dx ((1-x^(n+1))/(1-x)).

Now you can evaluate d/dx ((1-x^(n+1))/(1-x)) and set x:=1/(1+r).

Best wishes
Torsten.

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