Question
The function $$f\left( x \right) = \sin \frac{{\pi x}}{{n!}} - \cos \frac{{\pi x}}{{\left( {n + 1} \right)!}}$$ is :
A.
not periodic
B.
periodic, with period $$2\left( {n\,!} \right)$$
C.
periodic, with period $${\left( {n + 1} \right)}$$
D.
none of these
Answer :
none of these
Solution :
$$\sin \frac{{\pi x}}{{n!}}$$ has the period $$\frac{{2\pi }}{{\frac{\pi }{{n\,!}}}},$$ i.e., $$2\left( {n\,!} \right)$$ and $$\cos \frac{{\pi x}}{{\left( {n + 1} \right)!}}$$ has the period $$\frac{{2\pi }}{{\frac{\pi }{{\left( {n + 1} \right)!}}}},$$ i.e., $$2\left\{ {\left( {n + 1} \right)!} \right\}.$$ Their LCM $$ = 2\left\{ {\left( {n + 1} \right)!} \right\}.$$