I am doing an exercise from textbook which asks to compute Taylor series at $x = 0$ of $e^{1-\cos{x}}$ in summation notation?

My intuition was to expand function $e^x = \sum^{\infty}\_{k=0} \frac{x^k}{k!}$ and $\cos{x} = \sum^{\infty}\_{k=0}(-1)^k\frac{x^{2k}}{(2k)!}$ at ${x = 0}$, so I got the result: 
$\sum^{\infty}\_{k=0}\frac{1 - \sum^{\infty}\_{t=0}(-1)^t\frac{x^{2t}}{(2t)!}}{k!}$. 

The result is a form of 2 summations, thus I am not too sure whether it is correct. Could anyone please help to verify or give some comments?

Thanks for VVejalla's comment, the double summation result I got should be: $\sum^{\infty}\_{k=0}\frac{(1 - \sum^{\infty}\_{t=0}(-1)^t\frac{x^{2t}}{(2t)!})^k}{k!}$