Evaluate the sum $\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$ $$\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$$
I am having difficulty finding the function that represents this series. 
I have only found radius of convergence which is $(-\infty,+\infty)$ from the ratio test.

Let $f(x)$ be the function represented by your series. Then 
$$f(x) = \frac{1}{2}\sum_{n = 0}^\infty \frac{1 - (-1)^n}{n!}x^n$$
since $1 - (-1)^n$ is $0$ when $n$ is even and $2$ when $n$ is odd. Thus
$$f(x) =  \frac{1}{2}\sum_{n = 0}^\infty \frac{x^n}{n!} - \frac{1}{2}\sum_{n = 0}^\infty \frac{(-x)^n}{n!}, $$
which simplifies to 
$$f(x) = \frac{1}{2}e^x - \frac{1}{2}e^{-x}.$$
