Are there clever ways to evaluate this infinite series?

Here is an interesting infinite series. It would be great to see a method to evaluate it, if possible. I know it converges to a little less than 11/40

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{4^{k}+k!}$

I could not think of any good identities to start this.

Thanks a million to those who can show how to evaluate it.

Maybe even in general, $\displaystyle\sum_{k=1}^{\infty}\frac{1}{x^{k}+k!}$, where $x\geq 1$

The only approach I can think of is to narrow down the difference in the remainder term between the exact value of the series and the value of some partial sum that we use as estimate and judge how good or close our estimate is. Take the remainder term $R_n=\sum_{k=n+1}^\infty \frac{1}{4^k+k!}$ and $T_n=\sum_{k=n+1}^\infty \frac{1}{4^k}$ we know that $R_n\lt T_n\lt\int_{n+1}^\infty \frac{1}{4^k}dk $. The higher you choose your $n$ to be the lower the remainder or the error term. For example, $n=8$ partial sum is $s_7=0.2745411421$ and $R_7\lt 0.000011006$. So the partial sum is correct to atleast three or four decimal places. (This was too long for a comment.)