An exponential sum depending on a parameter Let $$\phi(k) = \frac{k^{\epsilon}}{\epsilon}$$ and consider  the function
$$ f(\epsilon) = \sum_{k=1}^{\infty} e^{-\phi(k)} $$ I believe $f(\epsilon)$ is finite when $\epsilon$ is small and positive. What happens to $f(\epsilon)$ as $\epsilon$ approaches zero from the right? Does it stay bounded or approach infinity? And if it approaches infinity, at what rate does it happen (i.e., as $1/\epsilon$, $1/\epsilon^2$, etc)? 
Update: Wolfram Alpha seems to strongly suggest the sum actually approaches zero as $\epsilon$ goes to zero. 
 A: $$
{\rm f}_{N}\left(\epsilon\right)
=
\sum_{k = 1}^{N}\exp\left(-{k^{\epsilon} \over \epsilon}\right)
=
N\sum_{k = 1}^{N}\exp\left(-{\left\lbrace N\left\lbrack k/N\right\rbrack\right\rbrace^{\epsilon} \over \epsilon}\right)\,{1 \over N}
\sim
N\int_{1/N}^{1}\exp\left(-N^{\epsilon}x^{\epsilon}/\epsilon\right)\,{\rm d}x
$$
when $N \gg 1$.
$t \equiv N^{\epsilon}x^{\epsilon}/\epsilon
\quad\Longrightarrow\quad
x = {\epsilon^{1/\epsilon} \over N}\,t^{1/\epsilon}$
$$
N\int_{1/N}^{1}\exp\left(-N^{\epsilon}x^{\epsilon}/\epsilon\right)\,{\rm d}x
=
N\int_{1/\epsilon}^{N^{\epsilon}/\epsilon}{\rm e}^{-t}\,
{\epsilon^{1/\epsilon} \over N}\,{1 \over \epsilon}\,t^{1/\epsilon - 1}
\,{\rm d}t
=
{\epsilon^{1/\epsilon} \over \epsilon}
\int_{1/\epsilon}^{N^{\epsilon}/\epsilon}{\rm e}^{-t}\,t^{1/\epsilon - 1}
\,{\rm d}t
$$
$$
{\large%
{\rm f}\left(\epsilon\right)
=
{\epsilon^{1/\epsilon} \over \epsilon}
\int_{1/\epsilon}^{\infty}{\rm e}^{-t}\,t^{1/\epsilon - 1}
\,{\rm d}t\,,\qquad \epsilon > 0}
$$
$$
\left.\vphantom{\LARGE A}{\rm f}\left(\epsilon\right)\right\vert_{\epsilon\ \gtrsim\ 0}
=
{\epsilon^{1/\epsilon} \over \epsilon}
\int_{1/\epsilon}^{\infty}{\rm e}^{-t}\,t^{1/\epsilon - 1}
\,{\rm d}t
\sim
\left.\vphantom{\LARGE A}
-{\epsilon^{1/\epsilon} \over \epsilon}\,{\rm e}^{-t}t^{1/\epsilon - 1}
\right\vert_{t\ =\ 1/\epsilon}^{t\ \to\ \infty}
=
{\epsilon^{1/\epsilon} \over \epsilon}\,{\rm e}^{-1/\epsilon}
{1 \over \epsilon^{1/\epsilon - 1}}
=
{\rm e}^{-1/\epsilon}
$$
$$
\begin{array}{|c|}\hline\\ \\
{\large\quad{\rm f}\left(\epsilon\right) \sim {\rm e}^{-1/\epsilon}
 \quad\mbox{when}\quad
 \epsilon \gtrsim 0\quad}
\\ \\
\hline
\end{array}
$$
