I want to solve this exercises but I can n't ,can you help me please?

I want to find sum of functions series , and 'i' here is an index.

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- Aug 4th 2008, 11:28 AM #1

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- Aug 4th 2008, 11:12 PM #2

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Well lets assume that for 1 you mean:

$\displaystyle s(n)=\sum_{i=1}^n \left( \frac{1}{x^{2i+1}}- \frac{1}{x^{2i-1}}\right)\ \ \ x \in (0,1]$

($\displaystyle x$ cannote be $\displaystyle 0$ as then the summands are undefined)

Then:

$\displaystyle

s(n)=\sum_{i=1}^n \frac{1}{x^{2i}} \left( \frac{1}{x}- x\right)

$

...... $\displaystyle =\left( \frac{1}{x}- x\right)\sum_{i=1}^n \frac{1}{x^{2i}}=\left( \frac{1}{x}- x \right) \left( \frac{1-x^{-2(n+1)}}{1-x^{-2}} \right),\ \ \ x \ne 1$

and $\displaystyle s(n)=0$ if $\displaystyle x=1$

RonL