1−((1/2))^(2k) +(((1.3)/(2.4)))^(2k) −(((1.3.5)/(2.4.6)))^(2k) +(((1.3.5.7)/(2.4.6.8)))^(2k) −.... Find the general form


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Question Number 136519 by Dwaipayan Shikari last updated on 22/Mar/21

$1 - {(\frac{1}{2})}^{2 k} + {(\frac{1.3}{2.4})}^{2 k} - {(\frac{1.3.5}{2.4.6})}^{2 k} + {(\frac{1.3.5.7}{2.4.6.8})}^{2 k} - . . . .$ $F i n d t h e g e n e r a l f o r m$
	Answered by mindispower last updated on 22/Mar/21

	$1 + \sum_{n ⩾ 1} \frac{{(- 1)}^{n} . \underset{m = 0}{\prod^{n - 1}} {(2 m + 1)}^{2 k}}{\underset{m = 0}{\prod^{n - 1}} . {(2 (m + 1))}^{2 k}}$ $= 1 + \sum_{n ⩾ 1} . {(- 1)}^{n} \frac{\underset{m = 0}{\prod^{n - 1}} {(m + \frac{1}{2})}^{2 k}}{\underset{m = 0}{\prod^{m = n - 1}} {(m + 1)}^{2 k}}$ $= 1 + \sum_{n ⩾ 1} \frac{{(\prod_{m ⩽ n - 1} (\frac{1}{2} + m))}^{2 k}}{{(\prod_{m ⩽ n - 1} (1 + m))}^{2 k - 1}} . \frac{{(- 1)}^{n}}{n!}$ $= 1 + \sum_{n ⩾ 1} . \frac{{(\frac{1}{2})}_{n}^{2 k} .}{{(1)}^{2 k - 1}} . \frac{{(- 1)}^{n}}{n!}$ $=_{2 k} F_{2 k - 1} \| \begin{matrix} (\frac{1}{2}), . . . . . . . . . (\frac{1}{2}) \\ ; (1), (1) . . . . . . . (1); [- 1] \end{matrix} \|$
	Commented by Dwaipayan Shikari last updated on 22/Mar/21

	$T h a n k s s i r!$
	Commented by mindispower last updated on 23/Mar/21

	$p l e a s u r$
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