First slide

Functions (XII)

Question

Period of the function $f (x)$ $= \sin x + \tan \frac{x}{2} + \sin \frac{x}{2^{2}} + \tan \frac{x}{2^{3}} + ... + \sin \frac{x}{2^{n} - 1} + \tan \frac{x}{2^{n}}$ is

Moderate

A

$π / 2$
B

$2^{n} π$
C

$2^{n - 1} π$
D

None

Solution

$f (x)$ = $\sin x + \tan \frac{x}{2} + \sin \frac{x}{2^{2}} + \tan \frac{x}{2^{3}} + ... + \sin \frac{x}{2^{n} - 1} + \tan \frac{x}{2^{n}}$
Each of the function on the right is periodic with periods $2 π, 2 π, 2^{2} π, 2^{2} π, ..., 2^{n} π, 2^{n} π$

Hence, $f (x)$ is periodic with period $2^{n} π$ .

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