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Chapter 10: Problem 130

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\epsilon\) there corresponds an integer \(N\) such that for all \(m\) and \(n\), $$m>N \text { and } n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon$$.

Short Answer

Expert verified
The sequence's convergence implies difference between terms past a certain index is less than any positive number.

Step by step solution

01

Understanding Convergent Sequences

A sequence \( \{a_n\} \) is said to be convergent if there exists a real number \( L \) such that for every positive number \( \epsilon \), there exists a positive integer \( N \) where for all \( n > N \), \( |a_n - L| < \epsilon \). This definition forms the basis of our proof.
02

Introduce the Condition for Convergence

By the definition of convergence, for any \( \epsilon > 0 \), there exists an integer \( N \) such that \( |a_n - L| < \frac{\epsilon}{2} \) for all \( n > N \). This helps us later in breaking down the problem into smaller pieces.
03

Apply the Triangle Inequality

For two terms \( a_n \) and \( a_m \) such that \( m, n > N \), we want to show \( |a_m - a_n| < \epsilon \). We can use the inequality \(|a_m - a_n| = |a_m - L + L - a_n| \leq |a_m - L| + |a_n - L|\).
04

Use the Definition of Convergence Again

Since by definition, \( |a_m - L| < \frac{\epsilon}{2} \) and \( |a_n - L| < \frac{\epsilon}{2} \) for \( m, n > N \), then \(|a_m - a_n| \leq |a_m - L| + |a_n - L| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon\).
05

Conclude the Proof

Finally, we established that for every \( \epsilon > 0 \), there exists an integer \( N \) such that if \( m, n > N \), then \( |a_m - a_n| < \epsilon \). Thus, the statement given in the problem is proven.

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Key Concepts

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

Definition of Convergence
In mathematics, a sequence is described as convergent if it approaches a specific value, called the limit, as the sequence progresses. Think of it as a path that gets nearer and nearer to a destination without ever actually missing the mark鈥攊t can get infinitely close! The formal definition states: a sequence \( \{a_n\} \) converges to a limit \( L \) if for every positive number \( \epsilon \), there exists a natural number \( N \) such that for all \( n > N \), the distance between \( a_n \) and \( L \) is less than \( \epsilon \):

\[ |a_n - L| < \epsilon \]

This implies that as you move further in the sequence, the values \( a_n \) edge ever closer to \( L \). It assures that, given any level of precision (\( \epsilon \)), there is a point in the sequence beyond which all subsequent terms will remain within that precision from \( L \). This idea is key to understanding the behavior of sequences.
Triangle Inequality
The triangle inequality is a crucial concept in mathematics that applies to sequences as well. It provides a way to compare the sum of distances with direct distances. For any real numbers \( x \) and \( y \), it states:

\[ |x + y| \leq |x| + |y| \]

This can be visualized using a triangle, where the sum of the lengths of any two sides of a triangle is always greater than or equal to the length of the third side. In the context of convergent sequences, you can think of it as showing that the difference between two terms \( a_m \) and \( a_n \) can be bounded by:

  • the distance from \( a_m \) to the limit \( L \)
  • plus the distance from \( a_n \) to the same limit \( L \)

This relationship greatens our understanding of distance and closeness in sequences and was essential in proving the convergence in the exercise.
Epsilon-Delta Definition
The epsilon-delta definition is a cornerstone of mathematical analysis, extensively used for defining limits, including those of sequences. It brings precision to mathematical statements about limits. In the context of a convergent sequence \( \{a_n\} \) approaching its limit \( L \), this definition states that:

For every \( \epsilon > 0 \) (an arbitrarily small positive number), there exists a corresponding integer \( N \) such that for all terms \( n \) in the sequence where \( n > N \), \( a_n \) will make the inequality hold:

\[ |a_n - L| < \epsilon \]

The utility of the epsilon-delta definition lies in its flexibility to any level of finite precision, ensuring that a function or sequence approaches a limit with as much certainty as desired. This property allows mathematicians to establish consistent rigor and precision in their concepts. It was used in the exercise to define the bounds within which sequence terms must lie for them to be considered convergent to the limit \( L \).

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Most popular questions from this chapter

Use Taylor's formula with \(n=2\) to find the quadratic approximation of \(f(x)=(1+x)^{k}\) at \(x=0\) ( \(k\) a constant). b. If \(k=3,\) for approximately what values of \(x\) in the interval [0,1] will the error in the quadratic approximation be less

a. Suppose that \(f(x)\) is differentiable for all \(x\) in [0,1] and that \(f(0)=0 .\) Define sequence \(\left\\{a_{n}\right\\}\) by the rule \(a_{n}=n f(1 / n)\) Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .\) Use the result in part (a) to find the limits of the following sequences \(\left\\{a_{n}\right\\}\) \(\mathbf{b} \cdot a_{n}=n \tan ^{-1} \frac{1}{n} \quad\) c. \(a_{n}=n\left(e^{1 / n}-1\right)\) d. \(a_{n}=n \ln \left(1+\frac{2}{n}\right)\)

According to the Alternating Series Estimation Theorem, how many terms of the Taylor series for \(\tan ^{-1} 1\) would you have to add to be sure of finding \(\pi / 4\) with an error of magnitude less than \(10^{-3} ?\) Give reasons for your answer.

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=\frac{\sin n}{n}\)

Estimate the error if \(P_{4}(x)=1+x+\left(x^{2} / 2\right)+\left(x^{3} / 6\right)+\left(x^{4} / 24\right)\) is used to estimate the value of \(e^{x}\) at \(x=1 / 2\).
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