Given is an odd function and defined everywhere,periodic with period and integrable on every interval.
Let .
Then prove that for every integer .
Given that is defined everywhere and integrable on every interval we can consider
being odd implies (easy to prove if you're not allowed to state)
Now you can see that a simple substitution should give you that .
Using the periodicity of f you should can show that any integral of f over an interval of length 2 has the same value. Therefore
Adding together these integrals gives g(2n) = 0
Hope this helps.