Is there a closed form solution to this infinite series?

x is a number between 0 and 1.

n can take any value between 1 and infinity.

1/n + x/(n+2) + x^2/(n+4) + x^3/(n+6) + ...................

Thanks!

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- October 30th 2010, 06:35 AMTitianIs there a closed form solution to this infinite series?
Is there a closed form solution to this infinite series?

x is a number between 0 and 1.

n can take any value between 1 and infinity.

1/n + x/(n+2) + x^2/(n+4) + x^3/(n+6) + ...................

Thanks! - October 30th 2010, 09:43 PMCaptainBlack
- November 26th 2010, 08:34 AMchisigma
Let's suppose that n is a 'natural number' so that we have a family of functions defined as...

(1)

First we set and then we start with n=1…

(1)

Now for n=2...

(2)

Now for n=3...

(3)

And now for n=4...

(4)

Observing (1), (2), (3) and (4) it seems that the general explicit expression for is...

(5)

Kind regards

P.S. : also the expressions like can be written as functions of x... - November 26th 2010, 09:35 AMCaptainBlack