Test the series for convergence or divergence.∞(−1)n nn3 + 4n = 1

The series you've given is an alternating series. The general form of an alternating series is ∑(-1)^n * a_n, where a_n > 0 for all n.

The series you've given is ∑(-1)^n * (n / (n^3 + 4n)), which simplifies to ∑(-1)^n * (1 / (n^2 + 4)).

To test for convergence, we can use the Alternating Series Test, which states that an alternating series converges if:

1. The terms a_n are decreasing. This is true for our series, as 1 / (n^2 + 4) decreases as n increases.

2. The limit as n approaches infinity of a_n is 0. This is also true for our series, as the limit as n approaches infinity of 1 / (n^2 + 4) is 0.

Therefore, by the Alternating Series Test, the series converges.