advanced-calculus-pove-that-n-0-1-n-2-2n-2n-n-cos-nx-cos-x-4-2cos-x-2-

Question Number 136581 by mnjuly1970 last updated on 23/Mar/21

\dots \dots . a d v a n c e d c a l c u l u s \dots .

p o v e t h a t ::

::: \underset{n = 0}{\sum^{\infty}} {\frac{{(- 1)}^{n}}{2^{2 n}} (\begin{matrix} 2 n \\ n \end{matrix}) c o s (n x)} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s (\frac{x}{2})}}

Answered by Dwaipayan Shikari last updated on 23/Mar/21

\frac{1}{2} \underset{n = 0}{\sum^{\infty}} (\begin{matrix} 2 n \\ n \end{matrix}) {(- \frac{e^{i x}}{2^{2}})}^{n} + (\begin{matrix} 2 n \\ n \end{matrix}) {(- \frac{e^{- i x}}{4})}^{n}

= \frac{1}{2} . \frac{1}{\sqrt{1 + e^{i x}}} + \frac{1}{2 \sqrt{1 + e^{- i x}}} = \frac{e^{\frac{i x}{2}} + 1}{2 \sqrt{e^{i x} + 1}} = \frac{1 + c o s (\frac{x}{2}) + i s i n (\frac{x}{2})}{2 \sqrt{1 + c o s (x) + i s i n (x)}}

= \frac{2 c o s (\frac{x}{4}) (c o s \frac{x}{4} + i s i n (\frac{x}{4}))}{2 \sqrt{2 c o s \frac{x}{2}} \sqrt{c o s \frac{x}{2} + i s i n \frac{x}{2}}} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s \frac{x}{2}}} . \frac{e^{i x / 4}}{e^{i x / 4}} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s \frac{x}{2}}}