Question

Find the period of $f\left(x\right)=\sin x+\tan \frac{x}{2}+\sin \frac{x}{2^2}+\tan \frac{x}{2^3}+....+\sin \frac{x}{2^{n-1}}+\tan \frac{x}{2^n}$.

• A. $2n\pi$
• B. $2^n\pi$
• C. $2^n\pi +\frac{\pi }{2}$
• D. It is not a periodic function

Answer 1

The correct option is B $2^n\pi$

We have,
Period of $(\sin x+\tan \frac{x}{2})$ is $2\pi$,
$(\sin \frac{x}{2^2}+\tan \frac{x}{2^3})$ is $2^3\pi$,
$(\sin \frac{x}{2^{n-1}}+\tan \frac{x}{2^n})$ is $2^n\pi$.
And so on for consecutive terms
LCM of $\{2\pi ,2^3\pi ,....,2^n\pi \}=2^n\pi$.
Hence, the period of $f\left(x\right)$ is $2^n\pi$.