does the sequence (((-1)^(n+1))*sqrt(n))/(n+ln(n)) converge?



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.

does the sequence (((-1)^(n+1))*sqrt(n))/(n+ln(n)) converge?

Explore the convergence of the sequence (((-1)^(n+1))*sqrt(n))/(n+ln(n)) using the Alternating Series Test. This detailed analysis includes the behavior of logarithmic terms and the use of asymptotic estimates, ideal for university-level calculus students.

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