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