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To determine the convergence or divergence of the series 

\[
\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{n^2 + 7},
\]

we should first examine the behavior of the terms as \( n \to \infty \). This series has alternating terms due to the \( (-1)^{n+1} \) factor, so it’s an **alternating series**.

### Step 1: Check the Limit of \( \frac{n}{n^2 + 7} \) as \( n \to \infty \)
Since the alternating series converges if:
1. The limit of the term goes to zero.
2. The terms \( \frac{n}{n^2 + 7} \) decrease in magnitude as \( n \) increases.

Let's evaluate:

\[
\lim_{n \to \infty} \frac{n}{n^2 + 7}.
\]

We can simplify this limit by dividing the numerator and the denominator by \( n^2 \):

\[
\lim_{n \to \infty} \frac{n}{n^2 + 7} = \lim_{n \to \infty} \frac{\frac{n}{n^2}}{\frac{n^2}{n^2} + \frac{7}{n^2}} = \lim_{n \to \infty} \frac{\frac{1}{n}}{1 + \frac{7}{n^2}}.
\]

As \( n \to \infty \), both \( \frac{1}{n} \) and \( \frac{7}{n^2} \) approach zero, so the limit is:

\[
\lim_{n \to \infty} \frac{n}{n^2 + 7} = 0.
\]

Since the limit is zero, this satisfies the first condition for convergence.

### Step 2: Apply the Alternating Series Test
The Alternating Series Test (Leibniz's test) states that an alternating series of the form \( \sum (-1)^n a_n \) converges if \( a_n \) is positive, decreasing, and approaches zero as \( n \to \infty \).

In this case, \( a_n = \frac{n}{n^2 + 7} \):
1. **Positive**: \( a_n > 0 \) for all \( n \).
2. **Decreasing**: As \( n \) increases, \( \frac{n}{n^2 + 7} \) decreases because the denominator grows faster than the numerator.
3. **Limit to zero**: As shown above, \( \lim_{n \to \infty} \frac{n}{n^2 + 7} = 0 \).

Thus, all conditions of the Alternating Series Test are satisfied, so the series **converges**.

Would you like more details on this solution, or do you have any questions?

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### Related Questions
1. How would the convergence of the series change if \( n^2 + 7 \) was replaced with \( n^2 - 7 \)?
2. What if the series was not alternating? How would that affect convergence?
3. How does the Alternating Series Test compare to the Ratio Test?
4. Could this series be analyzed with the Integral Test, and would it give the same result?
5. What is the significance of the limit being zero for the terms in an alternating series?

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**Tip:** For any series, checking if the terms approach zero as \( n \to \infty \) is a quick first step to test for potential convergence.

Determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{n^2 + 7} \). Evaluate \( \lim_{n \to \infty} \frac{n}{n^2 + 7} \).

Analyze the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{n^2 + 7} \) by applying the Alternating Series Test and evaluating the limit \( \lim_{n \to \infty} \frac{n}{n^2 + 7} = 0 \). Understand step-by-step the necessary conditions for convergence.

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