power series - Show that $sum_{k=0}^{infty}frac{x^{3k}}{(3k)!} = frac{1}{3} e^x + frac{2}{3} e^{-frac{x}{2}} cosleft(frac{sqrt{3}}{2} xright)$

Is there anyone who knows, and want to help, how to show that this is true $\sum_{k=0}^{\infty}\frac{x^{3k}}{(3k)!} = \frac{1}{3} e^x + \frac{2}{3} e^{-\frac{x}{2}} \cos\left(\frac{\sqrt{3}}{2} x\right)$ ?

I know that $\sum_{k=0}^{\infty}\frac{x^{k}}{k!}= e^x$, but how to use it I don't know. I can't find my lecture notes so it is like that.

I'm thankful for your help.

Answer

Outline: Differentiate three times and notice that the series satisfies $y'''=y$. Then solve that differential equation, and match with the conditions $y(0)=1$ and $y'(0)=y''(0)=0$, that you get by looking at the coefficients of the series.