Let's analyze the absolute convergence of the series:

$\sum_{n=2}^{\infty} \frac{(-1)^{n-1} (n+2)}{2^n + 5}$

Step 1: Consider the Absolute Value of the General Term

The general term of the series is:

$a_n = \frac{(-1)^{n-1} (n+2)}{2^n + 5}$

Taking the absolute value:

$|a_n| = \left|\frac{(-1)^{n-1} (n+2)}{2^n + 5}\right| = \frac{n+2}{2^n + 5}$

Step 2: Analyze the Series for Absolute Convergence

We now need to determine whether the series

$\sum_{n=2}^{\infty} \frac{n+2}{2^n + 5}$

converges.

Step 3: Apply the Comparison Test

Since $2^n$ dominates $5$ as $n$ becomes large, the term $\frac{n+2}{2^n + 5}$ behaves similarly to $\frac{n+2}{2^n}$.

This suggests comparing the series with a simpler one:

$\sum_{n=2}^{\infty} \frac{n}{2^n}$

We know that the series

$\sum_{n=1}^{\infty} \frac{n}{2^n}$

is a convergent series (this can be shown using the ratio test or by recognizing it as a sum of a geometric series after some manipulation).

Step 4: Apply the Ratio Test

We can also apply the Ratio Test directly to the series $\sum_{n=2}^{\infty} \frac{n+2}{2^n + 5}$ to ensure convergence.

Calculate the ratio of consecutive terms:

$\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} = \lim_{n \to \infty} \frac{\frac{n+3}{2^{n+1} + 5}}{\frac{n+2}{2^n + 5}}$

Simplify the expression:

$= \lim_{n \to \infty} \frac{n+3}{n+2} \cdot \frac{2^n + 5}{2^{n+1} + 5}$

For large $n$:

$\frac{n+3}{n+2} \approx 1$ $\frac{2^n + 5}{2 \cdot 2^n + 5} \approx \frac{1}{2}$

So the limit becomes:

$\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|} = \frac{1}{2}$

Since the limit is less than 1, by the Ratio Test, the series

$\sum_{n=2}^{\infty} \frac{n+2}{2^n + 5}$

converges.

Conclusion

Since the series of the absolute values converges, the original series converges absolutely.

Would you like more details on any step of this process?

Here are 5 related questions you might find useful:

1. How does absolute convergence relate to the convergence of a series with alternating signs?
2. Why does the series $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converge?
3. What other convergence tests can be used when the Ratio Test is inconclusive?
4. How can the Comparison Test be applied to more complex series?
5. What is the significance of the dominant term in the denominator when determining convergence?

Tip: When analyzing series, recognizing the dominant behavior of the terms (e.g., $2^n$ in the denominator) can significantly simplify convergence testing.