Answer
To prove that a function is odd, one must show that $$f(-x)=-f(x) \,.$$
In this question
$$f(x)=a_{2n+1}x^{2n+1}+ \cdots +a_3x^3+a_1x=\sum_{i=0}^na_{2i+1}x^{2i+1} \, .$$
One can easily find $f(-x)$ by noting the fact that the function $f(x)$ is the sum of terms containing odd powers of $x$ and for each $i=0, \cdots n$, one has $(-x)^{2i+1}=(-1)^{2i+1}x^{2i+1}=-x^{2i+1}$.
So $$f(-x)=-a_{2n+1}x^{2n+1}- \cdots -a_3x^3-a_1x=-\sum_{i=0}^na_{2i+1}x^{2i+1}=-f(x) \, .$$
Hence, $f$ is odd.
Work Step by Step
The details have been explained in the answer.
