Problem 1 Suppose \(\left\\{a_{i}\right\\}... [FREE SOLUTION]

Chapter 1: Problem 1

Suppose $\left\\{a_{i}\right\\}_{i \in I}$ and $\left\\{b_{i}\right\\}_{i \in J}$ are sequences in $\mathbb{Q}$ with $$\lim _{i \rightarrow \infty} a_{i}=\lim _{i \rightarrow \infty} b_{i}$$ Show that $a_{i} \sim b_{i}$.

Short Answer

Expert verified

The sequences $a_i$ and $b_i$ are asymptotically equivalent as $ \lim_{i \to \infty} \frac{a_i}{b_i} = 1 $.

Step by step solution

Understanding the Concept of Asymptotic Equivalence

Two sequences $ \{a_i\} $ and $ \{b_i\} $ are asymptotically equivalent, denoted $ a_i \sim b_i $, if $ \lim_{i \to \infty} \frac{a_i}{b_i} = 1 $. To prove this, we must analyze the behavior of $ \frac{a_i}{b_i} $ as $ i $ approaches infinity.

Applying Limit Definition

Given that $ \lim_{i \to \infty} a_i = L $ and $ \lim_{i \to \infty} b_i = L $ in $ \mathbb{Q} $, both sequences converge to the same limit $ L $. We need to show $ \lim_{i \to \infty} \frac{a_i}{b_i} = 1 $.

Manipulating Sequence Terms

Since $ a_i $ and $ b_i $ both approach $ L $, for any small $ \epsilon > 0 $, there exists an integer $ N $ such that for all $ i > N $, $|a_i - L| < \epsilon L $ and $|b_i - L| < \epsilon L$. Thus, both $ a_i $ and $ b_i $ are close enough to $ L $ when $ i $ is sufficiently large.

Exploring the Limit of the Ratio

Considering the ratio $ \frac{a_i}{b_i} $, we have:\[\frac{a_i}{b_i} = \frac{L + (a_i - L)}{L + (b_i - L)}\]As $ i \rightarrow \infty $, $ a_i - L \to 0 $ and $ b_i - L \to 0 $. Therefore, the ratio approximates:\[\frac{L + 0}{L + 0} = 1\]

Using the Limit Laws

By the Sandwich Theorem (Squeeze Theorem), since $ \lim_{i \to \infty} a_i = L $ and $ \lim_{i \to \infty} b_i = L $, their ratio approaches $ 1 $ as $ i \to \infty $. Thus, we can conclude that:\[\lim_{i \to \infty} \frac{a_i}{b_i} = 1\]Hence, $ a_i \sim b_i $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits

Limits are fundamental in calculus and mathematical analysis. They describe how a function or sequence behaves as it approaches a specific point or infinity. In our context, we are interested in the limits of sequences. When we say that the limit of sequence $ \{a_i\} $ as $ i \to \infty $ is $ L $, we mean that as $ i $ becomes very large, the terms $ a_i $ get closer and closer to $ L $.

The notation $ \lim_{i \to \infty} a_i = L $ denotes this behavior.
A limit is essential in determining the convergence of sequences, where a sequence converges if it approaches a specific limit.

Understanding limits helps us comprehend the concept of asymptotic equivalence, where two sequences are said to be asymptotically equivalent if the ratio of their terms approaches 1 as $ i $ goes to infinity.

Sequences

A sequence is an ordered list of numbers, each number in the list is called a term. Sequences can be finite or infinite. In mathematical analysis, we often deal with infinite sequences.

Sequences are usually written as $ \{a_i\} $ where $ i $ is the index running over the natural numbers.
Each term $ a_i $ is defined by a specific rule or formula.
The behavior of sequences as $ i \to \infty $ is usually studied to determine their limit or to understand their asymptotic properties.

In this exercise, $ \{a_i\} $ and $ \{b_i\} $ are sequences in $ \mathbb{Q} $ 鈥� the set of rational numbers. Both sequences are shown to converge to a common limit $ L $. This common behavior near their limit is crucial to showing that $ a_i \sim b_i $.

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a vital tool in calculus used to find the limit of a function or sequence by comparing it with two other functions or sequences whose limits are known and are equal.

The theorem states that if $ a_i \le c_i \le b_i $ for all sufficiently large $ i $, and $ \lim_{i \to \infty} a_i = \lim_{i \to \infty} b_i = L $, then $ \lim_{i \to \infty} c_i = L $ as well.
This theorem is especially useful when directly finding the limit of $ \{c_i\} $ is hard, but $ \{c_i\} $ is bound by two sequences that converge to the same limit.
In our exercise, the Squeeze Theorem helps confirm that $ \lim_{i \to \infty} \frac{a_i}{b_i} = 1 $, given that both sequences $ \{a_i\} $ and $ \{b_i\} $ converge to the same limit $ L $.

Using this theorem, we ensure that the asymptotic equivalence condition $ a_i \sim b_i $ holds true, tying together the concepts of limits and sequences seamlessly.

91影视

Suppose \(\left\\{a_{i}\right\\}_{i \in I}\) and \(\left\\{b_{i}\right\\}_{i \in J}\) are sequences in \(\mathbb{Q}\) with $$\lim _{i \rightarrow \infty} a_{i}=\lim _{i \rightarrow \infty} b_{i}$$ Show that \(a_{i} \sim b_{i}\).

Short Answer

Step by step solution

Understanding the Concept of Asymptotic Equivalence

Applying Limit Definition

Manipulating Sequence Terms

Exploring the Limit of the Ratio

Using the Limit Laws

Key Concepts

Limits

Sequences

Squeeze Theorem

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