Problem 133 - Thomas Calculus

Question

For a sequence \(\left\\{a_{n}\right\\}\) the terms of even index are denoted by \(a_{2 k}\) and the terms of odd index by \(a_{2 k+1} .\) Prove that if \(a_{2 k} \rightarrow L\) and \(a_{2 k+1} \rightarrow L,\) then \(a_{n} \rightarrow L\)

Step-by-Step Solution

Answer

The entire sequence converges to \( L \).

Step 1: Understand the Problem

We are given a sequence \( \{a_n\} \) where terms with even indices, \( a_{2k} \), converge to \( L \) and terms with odd indices, \( a_{2k+1} \), also converge to \( L \). We need to show that the entire sequence \( \{a_n\} \) converges to \( L \).

Step 2: Define the Goal

Our goal is to show that the sequence \( \{a_n\} \) converges to \( L \). By definition, for any given \( \varepsilon > 0 \), there must exist an integer \( N \) such that for all \( n \geq N \), \(|a_n - L| < \varepsilon\).

Step 3: Analyzing Even and Odd Indexed Terms

Given \( a_{2k} \rightarrow L \), for any \( \varepsilon > 0 \), there exists an integer \( N_1 \) such that for all \( k \geq N_1 \), \(|a_{2k} - L| < \varepsilon\). Similarly, since \( a_{2k+1} \rightarrow L \), there is an integer \( N_2 \) such that for all \( k \geq N_2 \), \(|a_{2k+1} - L| < \varepsilon\).

Step 4: Combine the Limits

Let \( N = \max(N_1, N_2) \). Then, for all \( n \geq 2N \), be it even or odd, the inequalities \(|a_{2k} - L| < \varepsilon\) and \(|a_{2k+1} - L| < \varepsilon\) will hold for all \( k \geq N \). Thus, for the entire sequence \( \{a_n\} \), \(|a_n - L| < \varepsilon\).

Step 5: State the Conclusion

Since \( |a_n - L| < \varepsilon \) for all \( n \geq 2N \), both even and odd indexed terms converge to \( L \), we have shown that \( a_n \rightarrow L \). Therefore, the entire sequence converges to \( L \).

Key Concepts

Even Indexed Terms

In the context of a sequence, even indexed terms are those elements which occur at positions that are multiples of two. For a sequence \( \{a_n\} \), such terms are represented by expressions like \( a_{2k} \), where \( k \) is a natural number. This means you start from the first term and skip one term in between, picking up every second term.

When analyzing convergence for even indexed terms separately, we focus on those terms and determine whether as \( k \) becomes very large, \( a_{2k} \) tends towards a particular value \( L \). If it does, we say that the even indexed terms of the sequence converge to \( L \). This is important because it implies a consistent behavior, despite overlooking the odd indexed terms. It simplifies the analysis by temporarily focusing on a sub-sequence of the original sequence.

• Even indexed terms provide insights into the convergence pattern of the sequence.
• Focus on whether these terms alone approximate towards a limit \( L \).

Odd Indexed Terms

Odd indexed terms refer to those elements within the sequence that fill in the gaps left by the even indexed terms. In a formulaic expression, odd terms can be denoted by \( a_{2k+1} \). This identifies the terms that come immediately after each even indexed term, creating their own sub-pattern within the sequence.

The concept of convergence for odd indexed terms works similarly to even indexed terms. We check whether, as the index increases, the sequence of odd indexed terms gravitates towards the same limit \( L \). This approach is crucial because it allows for an examination of the behavior of the entire sequence through its sub-sequences. It further solidifies the convergence argument by showing both even and odd terms approach the same limiting value.

• Odd indexed terms complement even terms, offering a full picture of the sequence's convergence.
• Verification of convergence towards the same limit ensures the entire sequence converges.

Epsilon-Delta Definition of Convergence

The concept of convergence of a sequence can be rigorously defined by the epsilon-delta (\( \varepsilon-\delta \)) definition. This is a fundamental idea in calculus and analysis, providing a formal criterion for establishing convergence.

When a sequence converges to a limit \( L \), it means for every positive number \( \varepsilon \), no matter how small, there exists a corresponding natural number \( N \) such that for all indices \( n \) greater than or equal to \( N \), the terms of the sequence satisfy the condition \(|a_n - L| < \varepsilon\). Essentially, beyond some point, all terms of the sequence are located within an \( \varepsilon \) distance from \( L \).

• The \( \varepsilon-\delta \) definition captures the notion of getting arbitrarily close to the limit.
• For sequences with both odd and even terms converging separately, the entire sequence satisfies the whole \( \varepsilon-\delta \) criterion.