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Chapter 10: Problem 29

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. \(a_{n}=\frac{1-2 n}{1+2 n}\)

Short Answer

Expert verified
The sequence converges to -1.

Step by step solution

01

Define the Sequence

We are given the sequence \(a_n = \frac{1 - 2n}{1 + 2n}\). To determine the convergence or divergence of this sequence, we need to find the limit as \(n\) approaches infinity.
02

Set Up the Limit

We need to compute \(\lim_{n \to \infty} \frac{1 - 2n}{1 + 2n}\). This will help us understand the behavior of the sequence as \(n\) grows larger.
03

Simplify the Expression

To simplify \(\frac{1 - 2n}{1 + 2n}\), divide both the numerator and the denominator by \(n\):\[\lim_{n \to \infty} \frac{\frac{1}{n} - 2}{\frac{1}{n} + 2}.\]As \(n\) approaches infinity, the terms \(\frac{1}{n}\) approach zero.
04

Evaluate the Limit

After simplifying, our expression becomes:\[\lim_{n \to \infty} \frac{0 - 2}{0 + 2} = \frac{-2}{2} = -1.\]Thus, the sequence \(a_n\) converges to -1.

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Key Concepts

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

Understanding the Limit of a Sequence
The limit of a sequence helps you understand the behavior of a sequence as the terms get larger and larger. In mathematical terms, finding the limit of a sequence involves determining what value the terms of the sequence approach as the number of terms grows infinitely large. To find a sequence's limit, follow these steps:
  • First, express the sequence in its general form, such as \(a_n = \frac{1 - 2n}{1 + 2n}\).
  • Next, calculate the limit as \(n\) approaches infinity, indicated as \(\lim_{n \to \infty} a_n\).
  • The goal is to simplify the expression to see what value the sequence approaches, if any.
In our specific example, the sequence simplifies to approach -1 as \(n\) gets very large, meaning it converges to the value of -1.
Infinity in Calculus
Infinity in calculus refers to a concept where numbers or expressions grow without bound. Imagine counting numbers that increase forever or a function that continuously rises. Calculus often deals with these infinite concepts to understand limits, integrals, and more.When it comes to sequences:
  • Checking if \(n\) approaches infinity helps determine the behavior of sequences or functions over large intervals.
  • Infinity is not a number but indicates an unbounded value.
  • In sequence analysis, we explore how terms behave as they go towards infinite values, to see if they converge or diverge.
In specific expressions like \(\frac{1 - 2n}{1 + 2n}\), infinity helps us realize that parts of the expression \(\left(\frac{1}{n}\right)\) become negligible, simplifying our calculations and interpretations.
Basics of Numerical Sequence Analysis
Numerical sequence analysis involves examining sequences to identify their behavior, whether they converge to a number or diverge. It involves various techniques to simplify and evaluate sequences effectively.Key points in analyzing sequences:
  • Formulate the mathematical expression for the sequence and recognize patterns.
  • Use division by \(n\) to ease complexity in expressions where terms grow rapidly.
  • Check the limit by finding the value as \(n\) approaches infinity.
Through these processes, we determine the nature of the sequence. In our exercise, simplifying the sequence \(a_n = \frac{1 - 2n}{1 + 2n}\) reveals its limit, showing convergence to -1, which is crucial in confirming its behavior over an infinite domain.

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Most popular questions from this chapter

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) \(1 /(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)

Prove that \(\lim _{n \rightarrow \infty} \sqrt[n]{n}=1\).

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. \(a_{n}=n \ln \left(1+\frac{2}{n}\right)\)

Determine if the sequence is monotonic and if it is bounded. \(a_{n}=\frac{3 n+1}{n+1}\)

Assume that each sequence converges and find its limit. \(a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}}\)
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