It was asked to find the sum of the AGP series $\left\{\dfrac{e^{\frac{1}{k}}+2 e^{\frac{2}{k}}+3 e^{\frac{3}{k}}+\cdots+k e^{\frac{k}{k}}}{k^2}\right\}$

I tried solving it with the formula for sum of $n^{th}$ terms of AGP series $\frac{a-[a+(n-1) d] r^n}{1-r}+\frac{d r\left(1-r^{n-1}\right)}{(1-r)^2}$ but my answer is not matching with the given solution.
The solution of this series was given as $-\dfrac{e^{\frac{1}{k}}(e-1)}{\left(e^{\frac{1}{k}}-1\right)^2}+\dfrac{k e^{1+ \frac{1}{k}}}{e^{\frac{1}{k}}-1}$

Thanks for any help.