Are they equal? Σ_(n=0) ^∞ ((sin(n+1))/(n+1)) and Σ


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Question Number 121496 by Lordose last updated on 08/Nov/20

$Are they equal ?$ $\underset{n = 0}{\sum^{\infty}} \frac{\sin (n + 1)}{n + 1} and \underset{n = 1}{\sum^{\infty}} \frac{\sin (n)}{n}$
Commented by Dwaipayan Shikari last updated on 08/Nov/20

$f i r s t s e r i e s$ $\frac{s i n 1}{1} + \frac{s i n 2}{2} + \frac{s i n 3}{3} + . . .$ $s e c o n d s e r i e s$ $\frac{s i n 1}{1} + \frac{s i n 2}{2} + . . . . . (t h e y a r e s a m e :)$
	Answered by TANMAY PANACEA last updated on 09/Nov/20

	$Q = \frac{s i n 1}{1} + \frac{s i n 2}{2} + . . + \frac{s i n n}{n} + . . \infty$ $P = \frac{c o s 1}{1} + \frac{c o s 2}{2} + . . + \frac{c o s n}{n} + . . \infty$ $P + i Q = \frac{e^{i}}{1} + \frac{e^{i 2}}{2} + \frac{e^{i 3}}{3} + . . + \frac{e^{i n}}{n} + . . \infty$ $P + i Q = t + \frac{t^{2}}{2} + \frac{t^{3}}{3} + . . + \frac{t^{n}}{n} + . . \infty$ $= - l n (1 - t)$ $= - l n (1 - e^{i})$ $= - l n (1 - c o s 1 - i s i n 1)$ $w e h a v e t o s e p a r a t e r e a l s n d i m a g i n a r y p a r t$
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