The convergence of the sequence $x^n$ depends on the value of $x$. Let's analyze the behavior of the sequence for different values of $x$:

1. $x = 1$:

   • The sequence is $1^n = 1$ for all $n$.
   • This sequence converges to 1.

2. $x = -1$:

   • The sequence is $(-1)^n$, which oscillates between 1 and -1.
   • This sequence does not converge because it does not settle on a single value.

3. $|x| < 1$:

   • When $|x| < 1$, $x^n$ gets smaller and smaller as $n$ increases.
   • The sequence converges to 0.

4. $|x| > 1$:

   • When $|x| > 1$, $x^n$ grows without bound as $n$ increases (if $x > 1$) or alternates with increasing magnitude (if $x < -1$).
   • The sequence diverges.

5. $x = 0$:

   • The sequence is $0^n = 0$ for all $n$.
   • This sequence converges to 0.

Summary of Convergence:

• The sequence $x^n$ converges if and only if $|x| \leq 1$ and $x \neq -1$. Specifically:
  • Converges to 0 if $|x| < 1$.
  • Converges to 1 if $x = 1$.
  • Does not converge if $x = -1$ or $|x| > 1$.

Would you like more details or have any further questions?

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Here are 8 related questions:

1. What is the convergence behavior of $x^n$ for complex values of $x$?
2. How does the sequence $\frac{x^n}{n!}$ behave in terms of convergence?
3. What is the convergence of the sequence $\sin(x^n)$ when $|x| < 1$?
4. Can the sequence $x^n$ be considered bounded? Under what conditions?
5. How does the limit superior and limit inferior apply to the sequence $x^n$?
6. What is the difference in convergence between a sequence and a series involving $x^n$?
7. How does the convergence of $x^n$ compare to $n^x$?
8. Can a divergent sequence like $x^n$ have a subsequence that converges?

Tip: Understanding the general behavior of sequences with respect to their growth and limit points can help you analyze more complex sequences.