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Determine whether the following series are absolutely convergent, conditionally convergent, or divergent

To determine whether the series 

\[
\sum_{n=1}^{\infty} \left( \frac{-n}{2n + 1} \right)^{5n}
\]

is absolutely convergent, conditionally convergent, or divergent, let's examine its behavior.

### Step 1: Examine Absolute Convergence
To check absolute convergence, we consider the series of the absolute values of the terms:

\[
\sum_{n=1}^{\infty} \left| \left( \frac{-n}{2n + 1} \right)^{5n} \right| = \sum_{n=1}^{\infty} \left( \frac{n}{2n + 1} \right)^{5n}.
\]

As \( n \to \infty \), the term \(\frac{n}{2n+1} \) approaches \( \frac{1}{2} \). Therefore, for large \( n \), we can approximate each term as:

\[
\left( \frac{n}{2n + 1} \right)^{5n} \approx \left( \frac{1}{2} \right)^{5n}.
\]

This forms a series that behaves similarly to a geometric series with ratio \( r = \frac{1}{2}^5 = \frac{1}{32} \), which is less than 1. Therefore, the series converges absolutely by comparison with a convergent geometric series.

### Conclusion
Since the series converges absolutely, we conclude that it is absolutely convergent.

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Would you like more details on any part of this solution, or do you have further questions?

### Related Questions
1. How would you determine the convergence of a series with factorial terms?
2. What is the difference between absolute and conditional convergence?
3. Can all series be classified as either absolutely or conditionally convergent?
4. How do geometric series tests work for series convergence?
5. What other tests could be used for convergence if the series was not geometric?

#### Tip
For complex series, applying the Ratio or Root Test can often simplify the analysis, especially when exponential or factorial terms are involved.

Determine whether the following series are absolutely convergent, conditionally convergent, or divergent: \( \sum_{n=1}^{\infty} \left( \frac{-n}{2n + 1} \right)^{5n} \)

Explore how to determine if the series \( \sum_{n=1}^{\infty} \left( \frac{-n}{2n + 1} \right)^{5n} \) is absolutely convergent, conditionally convergent, or divergent. This problem involves series convergence concepts, geometric series tests, and absolute convergence analysis.

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