Let $\displaystyle f(x)=\lim_{n\to\infty} \left(\frac{ax}{n}\left(\sum_{k=1}^n \frac{[k^2-e^{-x}+k-1]}{k(k+1)}\right)\right)+\lambda$
Find $f(x)$ if $[\,.]$ denotes G.I.F.
I know how to solve such type of questions (by splitting the integral at integers), but that, too, in case where argument of G.I.F. is quite simple. I've never encountered any problem as mentioned above. Here's the original question.
I just want to know how to tackle these type of questions.
Any hints are appreciated!
By the hint given by @Clayton, I've solved the problem myself and that was truly a promising start.
\begin{align} f(x)&=\lim_{n\to\infty} \frac{ax}{n}\left(\sum_{k=1}^n \frac{[k^2-e^{-x}+k-1]}{k(k+1)}\right)+\lambda\\ &=\lim_{n\to\infty} \frac{ax}{n}\left(\sum_{k=1}^n \frac{k^2+k-2}{k(k+1)}\right)+\lambda\\ &=\lim_{n\to\infty} \frac{ax}{n}\left(\sum_{k=1}^n 1-2\left(\frac{1}{k}-\frac{1}{k+1}\right)\right)+\lambda\\ &=\lim_{n\to\infty} \frac{ax}{n}\left(n+\frac{1}{n+1}-2\right)+\lambda\\ &=ax+\lambda. \end{align}