Σ_(n=1) ^∞ ((sin(2n+1)θ)/((2n+1)^2 ))=(θ/2)log(2)+sinθ((log(sinθ))/4) _2 F_1 ((1/2),(1/2);(3/2);sin^2 θ)+((sinθ)/(16)) _2 F_1 ((1/2),(1/2),(1/2);(3/2);(3/2);sin^2 θ) Prove or disprove 0<θ<π


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Question Number 136420 by Dwaipayan Shikari last updated on 21/Mar/21

$\underset{n = 1}{\sum^{\infty}} \frac{s i n (2 n + 1) θ}{{(2 n + 1)}^{2}} = \frac{θ}{2} l o g (2) + s i n θ \frac{l o g (s i n θ)}{4}_{2} F_{1} (\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; {s i n}^{2} θ) + \frac{s i n θ}{16}_{2} F_{1} (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \frac{3}{2}; {s i n}^{2} θ)$ $P r o v e o r d i s p r o v e 0 < θ < π$
	Answered by mindispower last updated on 21/Mar/21

	$θ = π, s i n (2 n + 1) θ = 0$ $l e f t = 0$ $r i g h t = \frac{π l n (2)}{2} + \frac{1}{4} + \frac{1}{16}$
	Commented by Dwaipayan Shikari last updated on 21/Mar/21

	$S o r r y i h a v e e d i t e d t h e q u e s t i o n$
	Commented by Dwaipayan Shikari last updated on 21/Mar/21

	$V a l i d f o r 0 < θ < π$
	Commented by mindispower last updated on 22/Mar/21

	$o k i w i l l t r y b u t t c h e k i t a g a i n$
	Answered by mindispower last updated on 22/Mar/21

	$\sum_{n ⩾ 0} \frac{s i n (2 n + 1) θ}{{(2 n + 1)}^{2}} \int = S = \sum_{n ⩾ 1} \frac{s i n (n θ)}{n^{2}} - \sum_{n ⩾ 1} \frac{s i n (2 θ)}{4 n^{2}}$ $r e c a l \sum_{n ⩾ 1} \frac{s i n (n θ)}{n^{2}} = {C l}_{2} (θ), c l a u s e n f u n c t i o n$ $w e g e t S = {C l}_{2} (θ) - \frac{{C l}_{2} (2 θ)}{4}$ $c l a u s e n i n t e g r a t i o n r e p r e s e n t a t i o n$ ${C l}_{2} (θ) = - 2 \int_{0}^{\frac{θ}{2}} l n (2 s i n (t)) d t$ $= u = s i n (t) = - 2 \int_{0}^{s i n (\frac{θ}{2})} \frac{l n (2 u)}{\sqrt{1 - u^{2}}} d u = {c l}_{2} (θ)$ ${(1 + x)}^{a} = 1 + \sum_{n ⩾ 1} \frac{\prod_{k ⩽ n - 1} (a - k)}{n!} . x^{n}$ ${c l}_{2} (θ) = - 2 \int_{0}^{s i n (\frac{θ}{2})} (1 + \sum_{n ⩾ 1} {(- 1)}^{n} \frac{\prod_{k ⩽ n - 1} (- \frac{1}{2} - k) u^{2 n}}{n!}) l n (2 u) d u$ $\int u^{2 n} l n (2 u) d u = \frac{u^{2 n + 1}}{2 n + 1} l n (2 u) - \frac{u^{2 n + 1}}{{(2 n + 1)}^{2}}$ $i t s e e m s g o o d s t a r t e w e c a n f i n_{p} F_{q}$ $s i n l o g (s i n . .) a l s o i t h i n k v e r r y l o n g w o r c k$ $= - 2 (1 + \sum_{n ⩾ 1} \frac{{(\frac{1}{2})}_{n}}{n!}) (\frac{{s i n}^{2 n + 1} (\frac{θ}{2}) l n (2 s i n (\frac{θ}{2}))}{(2 n + 1)} - \frac{{s i n}^{2 n + 1} (\frac{θ}{2})}{{(2 n + 1)}^{2}})$ $= - 2 (1 + \sum_{n ⩾ 1} \frac{{(\frac{1}{2})}_{n} {(\frac{1}{2})}_{n}}{2 . {(\frac{3}{2})}_{n}}$
	Commented by Dwaipayan Shikari last updated on 22/Mar/21

	$T h a n k s s i r! I w i l l p o s t m y w o r k l a t e r$
	Commented by mindispower last updated on 22/Mar/21

	$y o u a r e w e l c o m$
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