We are given a sequence $\{p_n\}_{n=1}^{\infty}$ defined as follows:

• $p_{2n+1} = 1$ for odd indices.
• $p_{2n} = \frac{1}{n}$ for even indices.

The question asks whether this sequence is a Cauchy sequence.

Cauchy Sequence Definition:

A sequence $\{p_n\}$ is called Cauchy if for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n > N$, the absolute difference $|p_n - p_m|$ is less than $\epsilon$. In other words, the terms of the sequence get arbitrarily close to each other as the sequence progresses.

Analysis of the Sequence:

1. For even indices $n=2k$, we have $p_{2k} = \frac{1}{k}$, which converges to 0 as $k \to \infty$.
2. For odd indices $n = 2k+1$, we have $p_{2k+1} = 1$, which is constant for all odd terms.

Thus, the sequence alternates between 1 (for odd indices) and values of $\frac{1}{n}$ (for even indices). Since $\frac{1}{n} \to 0$, the even-indexed terms approach 0, but the odd-indexed terms remain fixed at 1.

Checking the Cauchy Condition:

Consider the difference between terms of the sequence:

• If we take an odd index $p_{2k+1} = 1$ and an even index $p_{2k} = \frac{1}{k}$, the difference between these two terms is:

$|p_{2k+1} - p_{2k}| = |1 - \frac{1}{k}|$

As $k \to \infty$, this difference approaches 1, not 0.

Since the terms do not get arbitrarily close to each other (they maintain a difference of approximately 1 between odd and even terms), the sequence does not satisfy the condition for being Cauchy.

Conclusion:

The sequence $\{p_n\}$ is not a Cauchy sequence because the difference between the odd and even terms does not become arbitrarily small as $n$ increases.

Would you like further details or have any other questions?

Here are 5 related questions:

1. What is the formal definition of a convergent sequence?
2. How does a Cauchy sequence relate to convergence in a complete metric space?
3. Can a non-Cauchy sequence still be bounded?
4. What is an example of a sequence that is Cauchy but not convergent?
5. How does the behavior of even-indexed and odd-indexed terms affect the convergence of a sequence?

Tip: A sequence must get arbitrarily close to itself in order to be Cauchy. If the terms alternate or have large jumps, the sequence cannot be Cauchy.