What is the fundamental period of the function $$ f(x) = \sin x + \tan x + \tan\frac{x}{2} + \tan\frac{x}{4} + \tan\frac{x}{8} + \tan\frac{x}{16}~ .$$
I know that $16\pi$ is one period but how can I determine the fundamental period?
Can anyone please help me to find out it's fundamental period?
My friend was telling me that it's fundamental period will also be $~16\pi~$. Because $~16\pi~$ is the L.C.M of all periods of the periodic functions in the expression.
But I can not understand this argument because the well known function $~|\sin x | + |\cos x|~$ is a periodic function with period $~\frac{\pi}{2}~$ where as $~|\sin x |~$ and $~|\cos x|~$ are of period $~\pi ~$.
Best Answer
It is easy to check that $f(x + 16\pi) = f(x)$ for all $x \in \mathbb{R}$. Since $f$ is continuous on its domain (or even continuous as function $\mathbb{R} \to \mathbb{R}\cup\{\infty\}$) and non-constant, it follows that the fundamental period of $f$ is of the form $16\pi/n$ for some positive integer $n$.
Now if $n$ is a positive integer for which $16\pi/n$ is a period of $f$, then we must have $f(16\pi/n) = 0$.
If $n > 32$, then it is easy to see that each summand of $f(16\pi/n)$ is positive, and so, $f(16\pi/n) > 0$.
So it suffices to check that $f(16\pi/n) \neq 0$ for each $2 \leq n \leq 32$. This is the trickiest part, and to be honest, I do not see any clear argument for this. (Although we can easily remove $n = 2, 4, 8, 16, 32$ out of options, all the other values still deserve to be investigated.) But any CAS is capable of computing those values, and it turns that none of them are zero.
Therefore the only possible choice is $16\pi$.