interval of convergence

**WartonMorton** · May 31st 2010, 04:01 PM

Interval of Convergence of the following...

**tonio** · May 31st 2010, 06:17 PM

Originally Posted by WartonMorton

Interval of Convergence of the following...

1) This question doesn't belong here

2) It's practically impossible to decypher what you meant to write, in particular that weird middle symbol...??

Tonio

**chisigma** · May 31st 2010, 07:00 PM

Under the hypothesis that is...

$f(x)= \sum_{k=1}^{\infty} k a^{k} x^{k}$ (1)

... setting $a x = \xi$ we have...

$\sum_{k=1}^{\infty} k a^{k} x^{k}= \sum_{k=1}^{\infty} k \xi^{k}= \xi \sum_{k=1}^{\infty} k \xi^{k-1} = \xi \frac{d}{d \xi} \sum_{k=1}^{\infty} \xi^{k} =$

$= \xi \frac{d}{d \xi} \frac{\xi}{1-\xi} = \frac{\xi}{(1-\xi)^{2}}$ (2)

The series converges for $|\xi|<1$ ...

Kind regards

$\chi$ $\sigma$

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