Discuss the convergence of the series ∑
(
−
1
)
𝑛
−
1
𝑛
5
𝑛
+
1

The series in question is:

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

To discuss its convergence, we need to examine the nature of the series. Specifically, this is an **alternating series**, which means we can potentially apply the **Alternating Series Test (Leibniz criterion)**. However, given that the terms involve a more complex expression, we will need to investigate both the alternating nature and the general behavior of the terms.

### Steps for Testing Convergence:

1. **Alternating Series Test (Leibniz Criterion)**:
   For an alternating series \( \sum (-1)^{n-1} a_n \) to converge, the following two conditions must be satisfied:
   - \( a_n \) should be positive, decreasing, and tend to zero as \( n \to \infty \).

   Let's check if these conditions apply to \( \frac{n^5}{n+1} \):

   - **Positivity**: The terms \( \frac{n^5}{n+1} \) are positive for all \( n \geq 1 \).
   - **Monotonic Decrease**: To check whether \( a_n = \frac{n^5}{n+1} \) is decreasing, we would need to analyze the derivative of this function or compare consecutive terms.
   - **Limit to Zero**: 
     \[
     \lim_{n \to \infty} \frac{n^5}{n+1} = \lim_{n \to \infty} \frac{n^5}{n+1} = \lim_{n \to \infty} \frac{n^5}{n} = n^4 \quad \text {non-zero}= this means

Explore the convergence of the series ∑((-1)^(n-1) * n^5 / (n + 1)) through detailed analysis and application of the Alternating Series Test. This problem examines the alternating nature and behavior of terms in series convergence, suitable for advanced undergraduate study.

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