To determine if these sequences are Cauchy sequences, let's go through each problem step-by-step:

Problem 2: Determine if $x_n = (-1)^n + \frac{1}{n}$ is a Cauchy sequence.

A sequence $(x_n)$ is a Cauchy sequence if, for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n \geq N$, $|x_n - x_m| < \epsilon$.

1. Analyze the sequence behavior:

   • The term $(-1)^n$ oscillates between $1$ and $-1$ depending on whether $n$ is even or odd.
   • The term $\frac{1}{n}$ converges to $0$ as $n \to \infty$, but it does not affect the oscillation caused by $(-1)^n$.

2. Consider the distance between terms:

   • For even $n$, $x_n = 1 + \frac{1}{n}$.
   • For odd $n$, $x_n = -1 + \frac{1}{n}$.
   • Thus, for large $n$, the sequence still oscillates around $1$ and $-1$, meaning it does not settle down to a single value.

3. Conclusion:

   • Since $|x_n - x_m|$ does not get arbitrarily small for large $m$ and $n$ (due to oscillation between values near $1$ and $-1$), $x_n$ is not a Cauchy sequence.

Problem 3: Prove that $x_n = \sqrt{n+1} - \sqrt{n}$ is a Cauchy sequence.

1. Rewrite the expression: $x_n = \sqrt{n+1} - \sqrt{n} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}.$

2. Analyze the limit of $x_n$ as $n \to \infty$:

   • As $n \to \infty$, $\sqrt{n+1} + \sqrt{n} \to 2\sqrt{n}$, so $x_n \approx \frac{1}{2\sqrt{n}}$.
   • This shows that $x_n \to 0$ as $n \to \infty$.

3. Check if $x_n$ is Cauchy:

   • For large $n$, $|x_n - x_m|$ can be made arbitrarily small because both $x_n$ and $x_m$ get closer to $0$ as $n, m \to \infty$.
   • Specifically, for any $\epsilon > 0$, we can find an $N$ such that for all $n, m \geq N$, $|x_n - x_m| < \epsilon$.

4. Conclusion:

   • Since $|x_n - x_m|$ can be made arbitrarily small for large $n$ and $m$, $x_n$ is a Cauchy sequence.

Would you like more details on any step, or have any other questions?

Related Questions

1. What is the general definition of a Cauchy sequence?
2. How can we determine whether an oscillating sequence can still be Cauchy?
3. Why does the convergence of $x_n$ to zero indicate it's a Cauchy sequence in Problem 3?
4. What role does the limit $\sqrt{n+1} - \sqrt{n} \approx \frac{1}{2\sqrt{n}}$ play in determining the Cauchy property?
5. Can sequences that do not converge still be Cauchy sequences?

Tip

For oscillating sequences, checking the limit of their terms is not sufficient. You need to carefully examine if the distances between successive terms approach zero.