Are-they-equal-n-0-sin-n-1-n-1-and-n-1-sin-n-n-

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} + \dots

s e c o n d s e r i e s

\frac{s i n 1}{1} + \frac{s i n 2}{2} + \dots . . (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