To find the sum of the series $\sum_{n=1}^{\infty} a_n$, where we are given the partial sum $S_n = \frac{n^2 - 1}{5n^2 + 1}$, we need to examine the behavior of $S_n$ as $n \to \infty$.

Here’s the approach:

1. Calculate the Limit of $S_n$ as $n \to \infty$: Since $S_n$ represents the sum up to the $n$-th term, we can find the infinite sum by calculating the limit $\lim_{n \to \infty} S_n$.

   $\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{n^2 - 1}{5n^2 + 1}$

2. Simplify the Expression: To simplify, divide the numerator and the denominator by $n^2$:

   $\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{\frac{n^2}{n^2} - \frac{1}{n^2}}{\frac{5n^2}{n^2} + \frac{1}{n^2}} = \frac{1 - 0}{5 + 0} = \frac{1}{5}$

3. Conclusion: Therefore, the sum of the series $\sum_{n=1}^{\infty} a_n$ is $\frac{1}{5}$.

The answer is: $\frac{1}{5}$

Would you like further details on this process, or do you have any other questions?

Here are five related questions that might expand your understanding:

1. How do partial sums help in determining the convergence of a series?
2. What happens if the partial sum $S_n$ diverges as $n \to \infty$?
3. How do we analyze the convergence of series without explicit formulas for partial sums?
4. What other techniques can we use to evaluate sums of infinite series?
5. How does this approach differ from using tests like the ratio test or comparison test?

Tip: When dealing with series, always examine if a closed-form expression for the partial sum is available, as it often simplifies the convergence analysis.