Final answer:
The given alternating series diverges.
Step-by-step explanation:
To determine whether the alternating series ∑ₙ=1∞ (-1)ⁿ+1n/3 n⁴+1 converges or diverges, we can use the Alternating Series Test. This test states that if the terms of an alternating series decrease in absolute value and approach zero as n approaches infinity, then the series converges. Let's apply this test to the given series:
First, let's check if the terms of the series decrease in absolute value. The term (-1)ⁿ+1n/3 n⁴+1 can be rewritten as (-1)ⁿ+1/n^3(n⁴+1). Since the denominator n^3(n⁴+1) increases as n increases, the terms of the series do not decrease in absolute value.
Therefore, we can conclude that the given alternating series diverges.