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The given series is:

\[
\sum_{n=1}^{\infty} \frac{-5}{n}
\]

This series can be simplified to:

\[
-5 \sum_{n=1}^{\infty} \frac{1}{n}
\]

### Analysis

The series \(\sum_{n=1}^{\infty} \frac{1}{n}\) is a harmonic series, which is known to **diverge**. Therefore, multiplying it by a constant, \(-5\), does not change its convergence behavior.

### Conclusion

Since the harmonic series diverges, the given series \(\sum_{n=1}^{\infty} \frac{-5}{n}\) also **diverges**. It does not converge absolutely or conditionally.

Would you like a deeper explanation or have further questions on this topic?

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Here are some related questions to deepen understanding:

1. How does the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) differ from a convergent p-series?
2. What is absolute convergence, and why does it matter in determining convergence?
3. Can a series that includes negative terms ever converge?
4. How would the series behave if it were \(\sum_{n=1}^{\infty} \frac{-5}{n^2}\)?
5. What are some other common divergent series?

**Tip:** For testing convergence or divergence, the p-series and comparison tests are often very useful.

Does the series \(\sum_{n=1}^{\infty} \frac{-5}{n}\) converge absolutely, conditionally, or diverge?

Explore the convergence and divergence of the series \(\sum_{n=1}^{\infty} \frac{-5}{n}\). Understand why the series diverges based on the harmonic series, with step-by-step analysis.

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