De Wolfram Función Gamma de las ecuaciones (35)-(37) proporcionar
\begin{align}\tag{1}
\frac{1}{\Gamma(x)} = x + \gamma x^{2} + \sum_{k=3}^{\infty} a_{k} x^{k}
\end{align}
donde, $a_{1}=1$, $a_{2}=\gamma$,
\begin{align}\tag{2}
a_{n} = n a_{1} a_{n-1} - a_{2} a_{n-2} + \sum_{k=2}^{n} (-1)^{k} \zeta(k) \, a_{n-k}.
\end{align}
Ahora,
\begin{align}\tag{3}
\sum_{r=1}^{n} \frac{1}{\Gamma\left(\frac{1}{r}\right)} \approx H_{n} + \gamma H_{n,2} + \sum_{k=3}^{\infty} a_{k} H_{n,k},
\end{align}
donde $H_{n,r}$ son la generalización de la Armónica de los números dados por
\begin{align}\tag{4}
H_{n,r} = \sum_{s=1}^{n} \frac{1}{s^{r}}.
\end{align}
Ya que el límite es para valores grandes de $n$, $n \rightarrow \infty$, a continuación, utilizar la aproximación, Wolfram Armónico Número de Aproximaciones,
\begin{align}\tag{5}
H_{n,r} \approx \frac{(-1)^{r} \psi^{(r-1)}(1)}{(r-1)!} - \frac{1}{(r-1) \, n^{r-1} } \left( 1 + \mathcal{O}\left(\frac{1}{n}\right) \right)
\end{align}
para obtener
\begin{align}\tag{6}
\sum_{r=1}^{n} \frac{1}{\Gamma\left(\frac{1}{r}\right)} \approx H_{n} - \frac{\gamma}{n} + \sum_{k=2}^{\infty} \frac{(-1)^{k} a_{k}}{(k-1)!} \, \psi^{(k-1)}(1) + \mathcal{O}\left(\frac{1}{n^{2}} \right).
\end{align}
Desde entonces,
\begin{align}\tag{7}
- \ln \Gamma\left( \frac{1}{n} \right) \approx \frac{\gamma}{n} - \ln(n) + \mathcal{O}\left(\frac{1}{n^{2}}\right)
\end{align}
entonces
\begin{align}\tag{8}
\sum_{r=1}^{n} \frac{1}{\Gamma\left(\frac{1}{r}\right)} - \ln \Gamma\left( \frac{1}{n} \right) \approx H_{n} - \ln(n) + \sum_{k=2}^{\infty} \frac{(-1)^{k} a_{k}}{(k-1)!} \, \psi^{(k-1)}(1) + \mathcal{O}\left(\frac{1}{n^{2}} \right).
\end{align}
Tomando el límite cuando $n \rightarrow \infty$ y el uso de
\begin{align}
\lim_{n \rightarrow \infty} \left( H_{n} - \ln(n) \right) = \gamma
\end{align}
entonces
\begin{align}\tag{9}
\lim_{n \rightarrow \infty} \left[ \sum_{r=1}^{n} \frac{1}{\Gamma\left(\frac{1}{r}\right)} - \ln \Gamma\left( \frac{1}{n} \right) \right] = \sum_{k=1}^{\infty} \frac{(-1)^{k} a_{k}}{(k-1)!} \, \psi^{(k-1)}(1).
\end{align}
Desde
\begin{align}\tag{10}
\psi^{(m)}(x) = (-1)^{m+1} m! \zeta(m+1, x)
\end{align}
entonces
\begin{align}\tag{11}
\lim_{n \rightarrow \infty} \left[ \sum_{r=1}^{n} \frac{1}{\Gamma\left(\frac{1}{r}\right)} - \ln \Gamma\left( \frac{1}{n} \right) \right] = \gamma + \sum_{k=2}^{\infty} a_{k} \zeta(k).
\end{align}
