If $f\left(x\right)=\sin x+\tan \frac{1}{2}x+\sin \frac{1}{2^2}x$

                $+\tan \frac{1}{2^3}x+\mathrm{....}+\sin \frac{1}{2^{n-1}}x+\tan \frac{1}{2^n}x$  is a periodic function with period $k\pi$ , then $k=$

The functions

$\sin x,\sin \left(\frac{x}{2^2}\right),\sin \left(\frac{x}{2^4}\right),\mathrm{.....},\sin \left(\frac{x}{2^{n-1}}\right)$  are periodic with period

$2\pi ,2^3\pi ,2^5\pi ,\mathrm{...},2^n\pi$  respectively.

Also, the functions

$\tan \left(\frac{x}{2}\right),\tan \left(\frac{x}{2^3}\right),\tan \left(\frac{x}{2^5}\right),\mathrm{....},\tan \left(\frac{x}{2^n}\right)$

are periodic with period

$2\pi ,2^3\pi ,2^5\pi ,\mathrm{...},2^n\pi$ respectively.

Hence, the period of while function $f\left(x\right)$  is actually the LCM of

$2\pi ,2^3\pi ,2^5\pi ,\mathrm{...},2^n\pi$  such that period of $f\left(x\right)$  is $2^n\pi$

Given that $f\left(x\right)$  is periodic with period $k\pi$

$\Rightarrow 2^n\pi =k\pi$