First slide

Functions (XII)

Question

The function $f (x) = \sin (\frac{πx}{n!}) - \cos \frac{πx}{(n + 1)!}$ is

Moderate

A

non periodic
B

periodic with period 2(n!)
C

periodic with period 2 (n + 1)!
D

periodic with period n!

Solution

Since the period of $\sin x$ is $2 π$ so the period of $\sin \frac{πx}{n!}$ is $\frac{2 πn!}{π} = 2 (n!)$ The period of $\cos \frac{πx}{(n + 1)!}$ is
$2 (n + 1)!$ The period of $f (x) = 1 . c . m (2 n!), 2 (n + 1)! = 2 (n + 1)!$

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