The given series is:

$S = \sum_{n=2}^\infty \frac{5n + 1}{2n - 1}.$

Step 1: Analyze the series

This series sums from $n = 2$ to $\infty$, and the general term is:

$a_n = \frac{5n + 1}{2n - 1}.$

We need to check if the series converges and then attempt to find the sum if it does.

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Step 2: Convergence analysis

For a series to converge, the general term $a_n$ must approach 0 as $n \to \infty$. Let's compute the limit of $a_n$ as $n \to \infty$:

$a_n = \frac{5n + 1}{2n - 1} \approx \frac{5n}{2n} = \frac{5}{2} \quad (\text{as } n \to \infty).$

Since $a_n \not\to 0$ but instead approaches $\frac{5}{2}$, this series diverges.

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Conclusion

The series $\sum_{n=2}^\infty \frac{5n + 1}{2n - 1}$ does not converge, so its sum cannot be calculated—it diverges to infinity.

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Would you like to explore:

1. A detailed explanation of series convergence tests?
2. How to manipulate and simplify series terms?
3. Other examples of divergent series for comparison?
4. How to compute partial sums for this series?
5. What happens if the series starts from a higher index?

Tip: Always check the convergence of a series before attempting to calculate its sum.