by using fourier serie find the value of Σ


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Question Number 26751 by abdo imad last updated on 28/Dec/17

$b y u s i n g f o u r i e r s e r i e f i n d t h e v a l u e o f \sum_{n = 0}^{\propto} \frac{1}{{(2 n + 1)}^{2}}$
Commented by abdo imad last updated on 30/Dec/17

$w e k n o w t h a t / x / = \frac{π}{2} - \frac{4}{π} \sum_{p = 0}^{\propto} \frac{c o s (2 p + 1) x}{{(2 p + 1)}^{2}}$ $(t h e f u c t i o n i s 2 π p e r i o d i c e v e n) \Rightarrow l e t t a k e x = 0$ $0 = \frac{π}{2} - \frac{4}{π} \sum_{p = 0}^{\propto} \frac{1}{{(2 p + 1)}^{2}} \Rightarrow \sum_{p = 0}^{\propto} \frac{1}{{(2 p + 1)}^{2}} = \frac{π^{2}}{8} .$
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