taylor expansion - How to prove a series is greater than zero over an interval?

Show that the series $\sum\limits_{k=0}^\infty \frac{(-1)^k(x^{2k+1})}{(2k + 1)!}$ is greater than zero for $0

For a function to show something was greater than zero over an interval I would differentiate, show there was a maximum (or it is an increasing function) and then show that the two end points were also greater than zero. Since this function is the same as the sine function, I imagine there is an analogous way to do this for a series but can't think how. Can someone explain to me the method I should use, is differentiating again the right option? (Also I wish to do this particularly using series, and do not wish to show by the method I described above using the sine function). Thanks