Question

Series $\sqrt{1+\cos \alpha }+\sqrt{1+\cos 2\alpha }+\sqrt{1+\cos 3\alpha }+$.......... to n terms.

Answer 1

$\sqrt{1+\cos \alpha }+\sqrt{1+\cos 3\alpha }+.....+\sqrt{1+\cos 3\alpha }+....+\sqrt{1+\cos n\alpha }$

Using $2{\cos }^2\frac{x}{2}=1+\cos x$

$\Rightarrow \sqrt{2}[\cos \frac{\alpha }{2}+\cos \frac{2\alpha }{2}+\cos \frac{3\alpha }{2}+.....+\cos \frac{n\alpha }{2}]$........(1)

Using $\sum _{k=0}^{n-1}\cos (a+kd)=\frac{\sin \left(nd/2\right)}{\sin \left(d/2\right)}\cos \left(\frac{2a+(n-1)d}{2}\right)$

Here, $a=\frac{\alpha }{2},d=\frac{d}{2}$

$\therefore \left(1\right)\Rightarrow$

$\sqrt{2}\left[\frac{\sin \left(\frac{d\alpha }{4}\right)}{\sin \left(\frac{\alpha }{4}\right)}\cos \left(\frac{\alpha }{4}\right(n+1\left)\right)\right]$