Question

Find the upper and lower limits of the sequence \(\left\\{s_{n}\right\\}\) defined by $$s_{1}=0 ; \quad s_{2 m}=\frac{s_{2 m-1}}{2} ; \quad s_{2 m+1}=\frac{1}{2}+s_{2 m} .$$

Short answer

Answer: The upper and lower limits of the sequence are \(\frac{1}{2}\) and \(0\), respectively.

Step-by-step solution

Analyze the odd terms of the sequence.

First, let's list the odd terms of the sequence \(\left\\{s_{2m-1}\right\\}\) based on the given recursive relation: \(s_{1}=0\), \(s_{2m+1}=\frac{1}{2}+s_{2 m}\). \(s_1 = 0\) \(s_3 = \frac{1}{2} + s_2\) \(s_5 = \frac{1}{2} + s_4\) \(s_7 = \frac{1}{2} + s_6\) \(\cdots\) Observe that all odd terms depend on the previous even term. So, to find the upper and lower limits of the odd terms, we need to understand the behavior of the even terms.

Analyze the even terms of the sequence.

Now, let's list the even terms of the sequence \(\left\\{s_{2m}\right\\}\) based on the given recursive relation: \(s_{2m} = \frac{s_{2m-1}}{2}\). \(s_2 = \frac{s_1}{2} = 0\) \(s_4 = \frac{s_3}{2} = \frac{1}{4} + \frac{s_2}{2}\) \(s_6 = \frac{s_5}{2} = \frac{1}{4} + \frac{s_4}{2}\) \(s_8 = \frac{s_7}{2} = \frac{1}{4} + \frac{s_6}{2}\) \(\cdots\) Observe that all even terms have a similar structure: they all contain the constant term \(\frac{1}{4}\) and half of the previous even term. Also note that, since each even term is half of the previous one, the even terms of the sequence are decreasing towards zero.

Compute the odd and even term limits separately.

We can now compute the limit of each subsequence, starting with the odd terms. Odd Subsequence: As all odd terms depend on the previous even term, and since the even terms are decreasing to 0, the odd terms should be converging to \(\frac{1}{2} + 0 = \frac{1}{2}\). Therefore, the limit of odd terms is \(\frac{1}{2}\). Even Subsequence: Since the even terms are decreasing towards zero, the limit of even terms is 0.

Find the upper and lower limits of the sequence.

Now that we have the limits of both odd and even subsequences, we can determine the upper and lower limits of the original sequence \(\left\\{s_n\right\\}\). The upper limit (supremum) of the sequence is the maximum limit of both subsequences, which is \(\frac{1}{2}\). The lower limit (infimum) of the sequence is the minimum limit of both subsequences, which is 0. So, the upper and lower limits of the sequence are \(\frac{1}{2}\) and \(0\), respectively.

Key concepts

Sequences and Series

In the study of real analysis, sequences and series are foundational concepts.
A sequence is a list of numbers written in a specific order, following a certain rule. Each number in a sequence is called a term, and is often denoted as \(s_n\) where \(n\) is its position in the sequence.
In our example, the sequence \( \{s_n\} \) is defined through specific rules for even and odd terms.
Under these rules, to generate the sequence, we start at \(s_1 = 0\), and then use recursive formulas to find the subsequent terms. The sequence alternates between calculations for even and odd indices, creating a distinctive pattern of terms.
Key points for sequences include:

• Volume: Sequences go on indefinitely, providing unlimited terms following their rules.
• Pattern: Identifying the rule or formula is critical to understanding a sequence.
• Behavior: Analyzing how terms behave as \(n\) becomes very large can reveal important properties, such as convergence or divergence.

Understanding sequences helps one to delve deeper into more advanced topics, such as series, which involve summing the terms of a sequence.

Upper and Lower Limits

The upper and lower limits of a sequence are closely tied to the concepts of supremum and infimum, respectively.
These limits describe the bounds of the sequence as it proceeds towards infinity.
For the sequence \( \{s_n\} \), we first isolated the behavior of odd and even subsequences. Each showed distinct limiting behaviors:
- **Odd subsequence**: The odd terms, depending on the previous even ones, converge towards \(\frac{1}{2}\) because even terms trend towards zero.- **Even subsequence**: Due to each term halving the previous one, the even subsequence gradually declines to zero.
The process of determining the upper (supremum) and lower (infimum) limits is thus about understanding these subsequences:

• The upper limit of the entire sequence is determined by the greatest limit from either subsequence. In this case, \(\frac{1}{2}\) from the odd subsequence.
• The lower limit is the smallest limit from the subsequences, which is 0 from the even subsequence.

These values, \(\frac{1}{2}\) and 0 respectively, encapsulate the bounds within which the sequence's terms ultimately reside.

Subsequences

Subsequences are smaller sequences derived from the original sequence by selecting terms according to a specific rule.
In the case of \( \{s_n\} \), we consider even and odd subsequences separately to simplify the analysis.
Key points about subsequences include:

• Extraction: Subsequence terms maintain the order of the original sequence but may skip over terms.
• Convergence: Subsequence limits can be a powerful tool to infer the limits of the overall sequence.
• Relation to Limits: Often, the behavior of subsequences provides insights into the upper and lower limits of the original sequence.

By analyzing subsequences, especially those defined by even and odd indices, we can more easily compute limits and study convergence. For \( \{s_n\} \), simplifying into subsequences helped find that even terms go to 0, and odd terms to \(\frac{1}{2}\).
Thus, subsequences serve as a methodical way to segment and tackle complex sequence behaviors.