(1^2 /2)+(((1+(1/2))^2 )/2^2 )+(((1+(1/2)+(1/3))^2 )/2^3 )+(((1+(1/2)+(1/3)+(1/4))^2 )/2^4 )+...


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Question Number 134984 by Dwaipayan Shikari last updated on 09/Mar/21

$\frac{1^{2}}{2} + \frac{{(1 + \frac{1}{2})}^{2}}{2^{2}} + \frac{{(1 + \frac{1}{2} + \frac{1}{3})}^{2}}{2^{3}} + \frac{{(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4})}^{2}}{2^{4}} + . . .$ Find in a closed form\n
	Answered by 脩茂= last updated on 09/Mar/21

	$\underset{n = 1}{\sum^{\infty}} \frac{H_{n}^{2}}{2^{n}} = \frac{1}{1 - \frac{1}{2}} [{l n}^{2} (1 - \frac{1}{2}) + {L i}_{2} (\frac{1}{2})] = 2 {l n}^{2} 2 + 2 {L i}_{2} (\frac{1}{2}) = {l n}^{2} 2 + \frac{π^{2}}{6}$ $H_{n} : h a r m o n i c n u m b e r$
	Commented byDwaipayan Shikari last updated on 09/Mar/21

	$G r e a t s i r!$
	Answered by mnjuly1970 last updated on 09/Mar/21

	Commented byDwaipayan Shikari last updated on 09/Mar/21

	$G r e a t s i r! T h a n k i n g y o u!$
	Commented bymnjuly1970 last updated on 09/Mar/21

	$g r a t e f u l . . .$
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