Prove that $$\sum_{k=0}^{\infty}\left(1+\frac{k}{x}\right)^{-x}$$ is uniformly convergent on $x\in\left[a,\infty\right).$
According to the equality, $$\frac{x}{1+x}<\ln(1+x)$$ we have, \begin{eqnarray*} \left(1+\frac{k}{x}\right)^{-x} &=& \exp\left\{-x\ln\left(1+\frac{k}{x}\right)\right\} \\ &<& \exp\left\{-x\cdot\frac{k}{x+k}\right\} \end{eqnarray*}
How do I control the RHS in order to apply DCT here? Any suggestions?