Is there a closed form for $S=\sum_{n=1}^\infty\big(e-(1+1/n)^{n+1/2}\big)$?

Question

Is there a closed form for $S=\sum_{n=1}^\infty\left(e-\left(1+\dfrac{1}{n}\right)^{n+1/2}\right)$?

I think the 1/2 makes the sum converge.

The original question does not have the 1/2 and the resulting sum diverges.