To determine the absolute convergence of the series

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

we need to analyze the series by checking the absolute value of the terms.

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 need to check if the series

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

converges.

Step 3: Comparison Test or Ratio Test

To determine the convergence of the series, we can compare $|a_n|$ with a simpler series or apply the Ratio Test.

Notice that as $n$ becomes large, the term $2^n + 5$ is dominated by $2^n$, so:

$|a_n| \approx \frac{n+2}{2^n}$

We can compare this with the series:

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

Step 4: Apply the Ratio Test

The Ratio Test is a convenient method here. Consider 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 this expression:

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

As $n$ becomes large, $\frac{n+3}{n+2}$ approaches 1, and $\frac{2^n + 5}{2 \cdot 2^n + 5}$ approaches $\frac{1}{2}$.

Thus, the limit becomes:

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

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

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

converges.

Conclusion

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

Would you like to explore the concept of absolute convergence further or have any specific questions about this process?

Here are 5 related questions you might find interesting:

1. What is the difference between absolute convergence and conditional convergence?
2. How does the Ratio Test determine the convergence of a series?
3. Can a series converge conditionally but not absolutely?
4. What are other methods to test for convergence of a series besides the Ratio Test?
5. How does the Comparison Test work for determining the convergence of a series?

Tip: When applying the Ratio Test, always ensure to simplify the ratio of consecutive terms as much as possible to evaluate the limit accurately.