Power Series

**graticcio** · March 5th 2008, 05:56 AM

is there a closed-form for the following power series?
$\sum_{i = a}^{\infty} i \cdot x^{n \cdot i}$

p.s. i just know that
1. $\sum_{i = 0}^{\infty} x^{n \cdot i} = \frac{1}{1- x^n}$
2. $\sum_{i = 0}^{\infty} i \cdot x^{i} = \frac{x}{(1-x)^2}$

Thanks

**mr fantastic** · March 5th 2008, 08:04 PM

Originally Posted by graticcio

is there a closed-form for the following power series?
$\sum_{i = a}^{\infty} i \cdot x^{n \cdot i}$

p.s. i just know that
1. $\sum_{i = 0}^{\infty} x^{n \cdot i} = \frac{1}{1- x^n}$
2. $\sum_{i = 0}^{\infty} i \cdot x^{i} = \frac{x}{(1-x)^2}$

Thanks

Yes. Replace x with x^n in formula 2.

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