the convergence of the sequence 𝑎𝑛=𝑥1/𝑛*an*​=*x*1/*n* under two specific cases: when 𝑥>1 and when 0<𝑥<1.

Let's analyze the convergence of the sequence \( a_n = x^{\frac{1}{n}} \) under the two specific cases:

### 1. Case \( x > 1 \):

For \( x > 1 \), consider the sequence \( a_n = x^{\frac{1}{n}} \).

- **As \( n \) increases**, \( \frac{1}{n} \) becomes smaller and smaller, approaching 0.
- Therefore, \( x^{\frac{1}{n}} \) behaves like \( x^0 = 1 \) as \( n \) approaches infinity.
  
Thus, \( \lim_{n \to \infty} x^{\frac{1}{n}} = 1 \).

**Conclusion**: The sequence \( a_n = x^{\frac{1}{n}} \) converges to 1 when \( x > 1 \).

### 2. Case \( 0 < x < 1 \):

For \( 0 < x < 1 \), again consider the sequence \( a_n = x^{\frac{1}{n}} \).

- **As \( n \) increases**, \( \frac{1}{n} \) approaches 0, just as in the previous case.
- However, since \( 0 < x < 1 \), the sequence \( x^{\frac{1}{n}} \) will approach \( 1 \) from below.

For example, if \( x = \frac{1}{2} \), then \( \left(\frac{1}{2}\right)^{\frac{1}{n}} \) approaches 1 as \( n \) becomes large.

**Conclusion**: The sequence \( a_n = x^{\frac{1}{n}} \) also converges to 1 when \( 0 < x < 1 \).

### Summary:

- For both cases \( x > 1 \) and \( 0 < x < 1 \), the sequence \( a_n = x^{\frac{1}{n}} \) converges to 1 as \( n \) approaches infinity.

Would you like any further details or have additional questions?

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Here are 8 related questions:
1. How does the sequence \( a_n = \left(\frac{1}{n}\right)^n \) behave as \( n \) increases?
2. What is the convergence of the sequence \( a_n = e^{-\frac{n}{x}} \) for different values of \( x \)?
3. How does the sequence \( b_n = n^{\frac{1}{n}} \) compare in terms of convergence?
4. What happens to the sequence \( a_n = x^{n} \) when \( 0 < x < 1 \)?
5. How can the squeeze theorem be applied to analyze the convergence of sequences?
6. What is the convergence behavior of the sequence \( a_n = n \cdot x^{\frac{1}{n}} \) for \( x > 0 \)?
7. How does the sequence \( a_n = (1 + \frac{1}{n})^n \) converge?
8. Can you find the limit of the sequence \( a_n = \left(1 - \frac{1}{n}\right)^n \)?

**Tip:** Recognizing the general behavior of exponential sequences is crucial for understanding their convergence properties.

Explore the convergence behavior of the sequence 𝑎𝑛=𝑥1/𝑛 for cases when 𝑥 > 1 and 0 < 𝑥 < 1. Understand how the sequence approaches 1 as 𝑛 approaches infinity, backed by rigorous mathematical analysis and examples. Suitable for undergraduate students and anyone interested in limits and exponential sequences.

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