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Question Number 141489 by mathsuji last updated on 19/May/21

	Answered by qaz last updated on 19/May/21

	$s o m e o n e m a y b e a b l e t o d e a l w i t h$ $t h i s \frac{2}{\sqrt{π}} \int_{0}^{π / 2} Σ \frac{\cos^{2 n} x}{n^{2} + 1} d x . . . . .$
	Commented by mathsuji last updated on 19/May/21

	$s o l u t i o n S i r p l e a s e . . .$
	Answered by mindispower last updated on 19/May/21

	$Γ (n + \frac{1}{2}) = (n - \frac{1}{2}) . . . . . (\frac{1}{2}) Γ (\frac{1}{2}) = \sqrt{π} . {(\frac{1}{2})}_{n}$ $n^{2} + 1 = (n - i) (n + i)$ $Γ (n + 1) = n!$ $\sum_{n ⩾ 1} \frac{{(\frac{1}{2})}_{n} . \sqrt{π}}{(n + i) (n - i)}$ $(n + i) = \frac{{(i)}_{n}}{{(1 + i)}_{n}}$ $\Leftrightarrow \sum_{n ⩾ 1} \frac{\sqrt{π} . {(\frac{1}{2})}_{n} . {(1 + i)}_{n} {(1 - i)}_{n}}{{(i)}_{n} {(- i)}_{n}} . \frac{1}{n!}$ $= \sqrt{π} (_{3} F_{2} (\frac{1}{2}, (1 + i), 1 - i; i, - i; [1]) - 1)$ $W h e r_{3} F_{2} i s g e n e r e a l i z e d H y p e r g e o m e t r i c F u n c t i o n$
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