To use the Ratio Test on the given series, we need to evaluate the limit:

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Given the formula for $a_n$ is:

$a_n = \frac{(-1)^n n}{n^2 + 2n + 2}$

We can compute $\frac{a_{n+1}}{a_n}$ as follows:

$\frac{a_{n+1}}{a_n} = \frac{\frac{(-1)^{n+1} (n+1)}{(n+1)^2 + 2(n+1) + 2}}{\frac{(-1)^n n}{n^2 + 2n + 2}}$

Simplifying this expression:

$\frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1} (n+1)}{(-1)^n n} \times \frac{n^2 + 2n + 2}{(n+1)^2 + 2(n+1) + 2}$

The factor $(-1)^{n+1} / (-1)^n$ simplifies to $-1$, and we then have:

$\frac{a_{n+1}}{a_n} = - \frac{(n+1)}{n} \times \frac{n^2 + 2n + 2}{(n+1)^2 + 2(n+1) + 2}$

Now, let’s compute the limit of this ratio as $n \to \infty$. Simplifying the expressions for large $n$:

• $\frac{n+1}{n} \to 1$
• $\frac{n^2 + 2n + 2}{(n+1)^2 + 2(n+1) + 2} \to 1$

Thus, the limit becomes:

$L = \lim_{n \to \infty} \left( -1 \times 1 \right) = -1$

According to the Ratio Test:

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

Since $L = 1$, the Ratio Test does not provide a conclusion, and we must use another method to determine convergence.

Would you like help exploring other tests, or would you like more details on the Ratio Test?

Here are 5 questions related to the topic:

1. How do you apply the Root Test for convergence?
2. What is the meaning of conditional convergence and absolute convergence?
3. How do you analyze alternating series for convergence?
4. How can you use the Integral Test to determine series convergence?
5. What role does the behavior of the terms $a_n$ play in convergence analysis?

Tip: If the Ratio Test is inconclusive, consider checking the terms of the series for comparison with a known convergent or divergent series.