It is given that $$ L=\lim _{k \rightarrow \infty}\left\{\frac{e^{\frac{1}{k}}+2 e^{\frac{2}{k}}+3 e^{\frac{3}{k}}+\cdots+k e^{\frac{k}{k}}}{k^2}\right\} $$
I tried solving it but I am stuck on this, but it seems to be that numerator is an arithmetico-geometric sequence. Solution to this problem was given something like this:
$$s=-\dfrac{e^{\frac{1}{k}}(e-1)}{(e^{\frac{1}{k}}-1)^2}+\dfrac{ke^{1+\frac{1}{k}}}{e^{\frac{1}{k}}-1} \tag{1}\label{1}$$ where $s$ is sum of AGP series in the numerator. So $$\begin{align} L &= \displaystyle\lim_{k \to \infty} \dfrac{s}{k^2} \\ &= -(e+1)+e \tag{2}\label{2} \end{align}$$ So I am having difficulty in understanding \eqref{1} and \eqref{2} Any other aliter solution and help is appreciated.