What-is-the-sum-of-this-sequence-sin-2-x-sin-2-2x-sin-2-3x-sin-2-nx-

Question Number 7659 by Tawakalitu. last updated on 07/Sep/16

W h a t i s t h e s u m o f t h i s s e q u e n c e

{s i n}^{2} (x) + {s i n}^{2} (2 x) + {s i n}^{2} (3 x) + \dots + {s i n}^{2} (n x)

Commented by prakash jain last updated on 07/Sep/16

\sin^{2} n x = {(\frac{e^{i n x} - e^{- i n x}}{2 i})}^{2} = \frac{e^{2 i n x} - e^{- 2 i n x} - 2}{- 2}

= - \frac{e^{2 i n x}}{2} + \frac{e^{- 2 i n x}}{2} + 1

Given sum = \underset{j = 1}{\sum^{n}} \frac{e^{- 2 i j x}}{2} - \underset{j = 1}{\sum^{n}} \frac{e^{2 i j x}}{2} + \underset{j = 1}{\sum^{n}} 1

f i r s t 2 t e r m s a r e G P

= \frac{e^{- 2 i x} (e^{- 2 i n x} - 1)}{2 (e^{- 2 i x} - 1)} - \frac{e^{2 i x} (e^{2 i n x} - 1)}{2 (e^{2 i x} - 1)} - n