# Linear combination of periodic functions need not be periodic

While digging in MSE I found the following

Linear combination of periodic functions need not to be periodic. If $$f_1, f_2, f_3, \dots, f_k$$ are periodic functions with fundamental periods $$T_1, T_2, T_3, \dots, T_k$$ respectively, then $$\sum _{i=1} ^k a_i f_i$$ where $$a_1, a_2, \dots, a_k$$ are constants is periodic if $$\text{lcm } (T_1, T_2, \dots, T_k)$$ exists.

Does anyone has a proof available or a link to it?

• Just to get an intuition, consider $\sin x + \{x\}$, where $\{\cdot\}$ is the fractional part function. This is clearly not periodic because the period of $\sin x$ is $2\pi$ which is irrational, while the period of $\{x\}$ is 1. Aug 16, 2020 at 16:23
• I am aware of a proposition that a function of a sum of periodic functions is periodic if-f the ratio of the respective periods is rational , but somehow I got stumped in this one. Aug 16, 2020 at 16:41

Let $$M = \mathrm{lcm}(T_1, ..., T_k)$$. For each $$i$$, you have $$T_i \text{ }| \text{ } M$$, so because $$f_i$$ is $$T_i$$-periodic, it is also $$M-$$periodic.
So all the $$a_i f_i$$ are $$M-$$periodic, so it is not hard to see that there sum is also $$M-$$periodic.