Series

The sum series is given by:


Similarly, we have the following series:


The general term is given by:


Now, calculating :


The sum is given by:


Using the known sum formulas for squares and sums of natural numbers:


Now, we calculate :

To solve the limit , we begin by considering the general term of the series:

.

The function in the numerator is a polynomial of degree 3. Because this is much smaller than a factorial , the series converges quickly.

We will use asymptotic analysis to simplify and approximate the sum. Consider rewriting:

.

Comparing the asymptotic behavior:

.

However, this lacks precision. Let's use the Taylor expansion strategy near large :

yielding the functional series rapidly converging.

The factorial growth suggests diminishing terms in distribution, aligning mass weights over limits.

Thus, focusing components finds sum aggregate:

Hence, the value of the limit is .