We need to determine whether the sequence $a_n = \frac{(-1)^{n+1} \sqrt{n}}{n + \ln(n)}$ converges as $n \to \infty$. Let’s carefully analyze it.

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Step 1: Check for Alternating Behavior

The factor $(-1)^{n+1}$ causes the sequence to alternate in sign. This suggests that the sequence might converge conditionally, or diverge if the positive and negative terms do not balance properly.

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Step 2: Behavior of the Absolute Value $|a_n|$

To apply convergence tests, let’s first look at the absolute value of the sequence:

$|a_n| = \frac{\sqrt{n}}{n + \ln(n)}.$

We need to analyze whether this absolute value goes to 0 as $n \to \infty$.

• As $n \to \infty$, the $n$-term in the denominator dominates $\ln(n)$ because $n$ grows much faster than $\ln(n)$. Thus, the denominator behaves asymptotically like $n$.

• Therefore, the sequence behaves like:

  $|a_n| \approx \frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}}.$

• As $n \to \infty$, $\frac{1}{\sqrt{n}} \to 0$. This suggests that the absolute values of the terms shrink to 0.

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Step 3: Alternating Series Test

Since the sequence alternates in sign, we can apply the Alternating Series Test (Leibniz’s test). The test requires:

1. The absolute value $|a_n|$ should decrease monotonically (i.e., $|a_{n+1}| < |a_n|$ for large $n$).
2. $|a_n| \to 0$ as $n \to \infty$.

• From Step 2, we saw that $|a_n| \approx \frac{1}{\sqrt{n}}$, which decreases monotonically and approaches 0 as $n \to \infty$.

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Step 4: Conclusion

Since the terms $|a_n|$ decrease monotonically to 0, the Alternating Series Test guarantees that the sequence $a_n = \frac{(-1)^{n+1} \sqrt{n}}{n + \ln(n)}$ converges.

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Would you like further details or additional questions about convergence or other related topics?

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Five Related Questions:

1. What is the Alternating Series Test, and when can it be applied?
2. How does the behavior of logarithmic terms compare with polynomials for large $n$?
3. What is conditional convergence, and how is it different from absolute convergence?
4. Can a sequence that passes the Alternating Series Test fail to be absolutely convergent?
5. How do growth rates like $\ln(n)$, $n^p$, and $e^n$ compare as $n \to \infty$?

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Tip:

When working with sequences involving alternating signs, always try to isolate the absolute value to simplify the analysis and then apply the Alternating Series Test if needed.