Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{n^{2}+1} $$

Short answer

The series does not converge.

Step-by-step solution

Step 1. Identify the series type

The given series is of the form \((-1)^n a_n\), where \(a_n = \frac{n^2}{n^2 + 1}\). Hence, it is classified as an alternating series.

Step 2. Apply the Alternating Series Test

The series \(\sum_{n=1}^{\infty} (-1)^n a_n\) converges if both of these conditions are met: 1) \(\lim_{n\rightarrow\infty}a_n = 0\), and 2) the sequence \(a_n\) is decreasing for all \(n\). Checking these conditions:

Step 3. Check for limit condition

Checking if \(\lim_{n\rightarrow\infty} \frac{n^2}{n^2 + 1} = 0\) yields that the limit is indeed 1, not 0. Therefore, the first condition of the Alternating Series Test is not satisfied.

Step 4. Final Reasoning

Since the first condition of the Alternating Series Test is not met, we can conclude at this point, without checking for the second condition. Therefore, the alternating series does not converge.