Question #41521

This question tests the student's ability to handle limits of sums where terms resemble an Arithmetico-Geometric Progression (AGP).

The trap lies in using the Telescopic Series property (or the infinite AGP formula, which is derived telescopically) to simplify the sum before passing to the limit $n \to \infty$.

If a student blindly applies the infinite sum formula $\sum_{k=1}^\infty k^2 r^k = \frac{r(1+r)}{(1-r)^3}$ with $r = 1 - 1/n$, they will incorrectly obtain a limit of 2.

This fails because the ratio $r$ depends on $n$ and approaches the convergence boundary $r=1$, meaning the telescopic boundary terms at $k=n$ (which evaluate to $-5/e$) cannot be ignored.

The correct approach is to recognize the expression as a Riemann sum.