To solve the last question by substituting a suitable $n$ in the equation, let’s rewrite the given series and analyze its structure:

We are tasked with evaluating: $\sum_{n=3}^\infty \frac{5 + 7n - 3n^2}{[n(n+1)]^2}.$

Step 1: Understand the structure

The denominator $[n(n+1)]^2$ is structured as $(n^2 + n)^2$. This structure suggests we focus on simplifying the numerator $5 + 7n - 3n^2$ into a form that can pair naturally with this denominator.

Step 2: Test substitutions

Instead of directly applying partial fractions, use the hint from the denominators in the earlier parts and structure the numerator to simplify cancellation.

Rewrite the general term

Let’s test substituting $n$ from earlier terms: For $n=3$, substitute into (\frac{5 + 7n - 3n^2}{[n(n+1)]^2}\