Final answer:
To test the convergence or divergence of the given series, we first find the general term, bn, which is (-1)^n * (2n + 2) / (n + 3). Then we split the series into two separate series and evaluate their limits as n approaches infinity. The limits are not the same, indicating that the given series diverges.
Step-by-step explanation:
To test the convergence or divergence of the series, we need to find the limit of the terms of the series as n approaches infinity.
First, let's find the general term, bn. The numerator of each term alternates between negative and positive even numbers, starting with -2 and increasing by 2 in each term. The denominator of each term is n + 3.
So, bn = (-1)^n * (2n + 2) / (n + 3).
Now, let's evaluate the limit of bn as n approaches infinity. Since the numerator has alternating signs, we can split the series into the sum of two separate series: Σ((-1)^n * 2n) / (n + 3) and Σ((-1)^n * 2) / (n + 3).
The limit of Σ((-1)^n * 2n) / (n + 3) as n approaches infinity is 2.
The limit of Σ((-1)^n * 2) / (n + 3) as n approaches infinity is 0.
Since the limit of the two series is not the same, the given series diverges.