Infinite Limit?

$\begin{aligned} L & = \frac 1{e^{1/2}}+ \frac 1{e^{2/3}}+ \frac 1{e^{3/4}}+...\\ L & > \frac 1e + \frac 1e+ \frac 1e+...\end{aligned}$

Since the RHS diverges and $L>$ RHS, $L$ diverges, $\implies L \to \boxed{\infty}$

$\large{\displaystyle \lim_{n\to \infty} (\frac{1}{e^{1-1/2}}+\frac{1}{e^{1-1/3}}+...+\frac{1}{e^{1-1/n}})}$

$\large{\frac{1}{e} \displaystyle \lim_{n\to \infty} \sum_{r=2}^{n} e^{\frac{1}{r}}}$

Now

$\large{\displaystyle \lim_{n\to \infty} e^{\frac{1}{n}}=1}$

and $e^{1/2}=1~point ~something\quad e^{1/3}=1~ point~something$ and so on

$\frac{1}{e}(1.abc+1.pqr+1.xyz+.....infinite~terms)$

Answer is $\infty$