Estimation of the absolute value of an infinite sum by its coefficient

I got this problem from a complex funtion, but it now has little to do with the complex analysis.$$\Theta(s)=\sum_{n=0}^{\infty}a_{2n}s^{2n}$$

I already have:$$c_1^n\frac{(\ln n)^{2n}}{(2n)!}\leq |a_{2n}|\leq c_2^n\frac{(\ln 2n)^{2n}}{(2n)!}$$

I want to confirm:$$|\Theta(s)|=O(s^s)$$

I'm not familiar with the estimation of the series, could anyone provide any solutions or thoughts?