1. **State the problem:** Determine if the infinite series $$\sum_{n=1}^\infty \frac{5n}{5n + 4n^{-10000}}$$ converges. 2. **Simplify the general term:** The term inside the sum is $$\frac{5n}{5n + 4n^{-10000}}$$. 3. **Rewrite the denominator:** $$5n + 4n^{-10000} = 5n + \frac{4}{n^{10000}}$$. 4. **Divide numerator and denominator by $n$ to analyze behavior as $n \to \infty$:** $$\frac{5n}{5n + 4n^{-10000}} = \frac{5n}{5n + 4n^{-10000}} \cdot \frac{\cancel{\frac{1}{n}}}{\cancel{\frac{1}{n}}} = \frac{5}{5 + 4n^{-10001}}$$ 5. **Evaluate the limit of the term as $n \to \infty$:** Since $n^{-10001} \to 0$, $$\lim_{n \to \infty} \frac{5}{5 + 4n^{-10001}} = \frac{5}{5 + 0} = 1$$ 6. **Apply the Divergence Test:** If the limit of the terms of a series does not approach zero, the series diverges. Here, the limit is 1, not zero. 7. **Conclusion:** The series $$\sum_{n=1}^\infty \frac{5n}{5n + 4n^{-10000}}$$ diverges because its terms do not approach zero.