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Question Number 29844 by abdo imad last updated on 12/Feb/18

l e t g i v e f_{α} (t) = c o s (α t) 2 π p e r i o d i c w i t h t \in [- π, π] a n d

α \in R - Z

1) d e v e l o p p f_{α} a t f o u r i e r s e r i e a n d p r o v e t h a t

c o t a n (α π) = \frac{1}{α π} + \sum_{n = 1}^{\infty} \frac{2 α}{π (α^{2} - n^{2})}

2) l e t x \in] 0, π [a n t g (t) = c o t a n t - \frac{1}{t} i f t \in] 0, x] a n d g (0) = 0

p r o v e t h a t g i s c o n t i n u e i n [0, x] a n d f i n d \int_{0}^{x} g (t) d t

3) p r o v e t h a t \forall t \in [0, x] g (t) = 2 t \sum_{n = 1}^{\infty} \frac{1}{t^{2} - n^{2} π^{24}}

4) c h o w t h a t \prod_{n = 1}^{\infty} (1 - \frac{x^{2}}{n^{2} π^{2}}) = \frac{s i n x}{x} a n d f o r x \in] - π, π [

s i n x = x \prod_{n = 1}^{\infty} (1 - \frac{x^{2}}{n^{2} π^{2}}) .