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Chapter 2: Problem 2

For each of the following sequences \(\left\\{a_{n}\right\\}\), draw its sequence diagram and show that \(\left\\{a_{n}\right\\}\) converges to \(\ell\) by considering \(a_{n}-\ell\) : (a) \(a_{n}=\frac{n^{2}-1}{n^{2}+1}, \ell=1\); (b) \(a_{n}=\frac{n^{3}+(-1)^{n}}{2 n^{3}}, \ell=\frac{1}{2}\).

Short Answer

Expert verified
Both sequences converge: (a) to 1 and (b) to 0.5 as given.

Step by step solution

01

Understanding the Sequence (a)

Given the sequence \(a_n = \frac{n^2-1}{n^2+1}\), we need to analyze its convergence as \(n\) approaches infinity. We suspect it converges to \(\ell = 1\).
02

Simplifying the Expression for (a)

Rewrite \(a_n\) as \(a_n = \frac{n^2-1}{n^2+1} = \frac{n^2+1 - 2}{n^2+1} = 1 - \frac{2}{n^2+1}\). This shows that \(a_n\) approaches 1 as \(n\) becomes very large.
03

Showing Convergence for (a)

To show convergence, consider \(a_n - \ell = \left(1 - \frac{2}{n^2+1}\right) - 1 = -\frac{2}{n^2+1}\). As \(n \to \infty\), \(\frac{2}{n^2+1} \to 0\), hence \(a_n - 1 \to 0\) proving convergence to \(1\).
04

Understanding the Sequence (b)

Given the sequence \(a_n = \frac{n^3 + (-1)^n}{2n^3}\), we need to analyze its convergence as \(n\) approaches infinity. We suspect it converges to \(\ell = \frac{1}{2}\).
05

Simplifying the Expression for (b)

Simplify \(a_n\) as \(a_n = \frac{n^3}{2n^3} + \frac{(-1)^n}{2n^3} = \frac{1}{2} + \frac{(-1)^n}{2n^3}\). This expresses \(a_n\) in terms of \(\frac{1}{2}\) plus a diminishing term.
06

Showing Convergence for (b)

To show convergence, consider \(a_n - \ell = \frac{1}{2} + \frac{(-1)^n}{2n^3} - \frac{1}{2} = \frac{(-1)^n}{2n^3}\). As \(n \to \infty\), \(\frac{(-1)^n}{2n^3} \to 0\), hence \(a_n - \frac{1}{2} \to 0\) proving convergence to \(\frac{1}{2}\).

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Key Concepts

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

Sequence Diagram
A sequence diagram is a helpful visual tool used to understand the behavior of sequences as the index \(n\) increases. When analyzing sequences, especially concerning their convergence, drawing a sequence diagram can make abstract mathematics more tangible.

  • Each term \(a_n\) is plotted on a graph where the horizontal axis represents the index \(n\) and the vertical axis represents the value of the sequence term, \(a_n\).
  • As \(n\) increases, the sequence diagram shows how the terms of the sequence approach the limit \(\ell\).
  • For example, in sequence (a) \(a_n = \frac{n^2-1}{n^2+1}\), plotting each value will show the terms approaching the line \(y=1\). Similarly, for (b) \(a_n = \frac{n^3+(-1)^n}{2n^3}\), the diagram will reveal the approach to \(y=\frac{1}{2}\).
Sequence diagrams offer an intuitive illustration, helping you visually confirm the convergence of a sequence to a particular limit.
Convergence Proof
In mathematics, proving the convergence of a sequence requires demonstrating that the terms eventually get arbitrarily close to a particular value, known as the limit. This is usually done through a formal limit definition or by using algebraic manipulations.

  • The key is to express \(a_n - \ell\) where \(a_n\) is the sequence term, and \(\ell\) is its claimed limit.
  • For example, consider \(a_n = \frac{n^2-1}{n^2+1}\) with \(\ell = 1\). We simplify to show \(a_n - 1 = -\frac{2}{n^2+1}\).
  • As \(n\) grows, \(-\frac{2}{n^2+1}\) approaches 0, confirming the sequence converges to \(1\).
Convergence proofs rely on showing that the distance between the sequence terms and the limit diminishes to zero as \(n\) gets infinitely large.
Limit of a Sequence
The limit of a sequence \(\ell\) is the value that the sequence terms \(a_n\) approach as \(n\) becomes very large. Understanding limits is fundamental to calculus and helps in defining functions' behavior at infinity.

  • Formally, \(\lim_{n \to \infty} a_n = \ell\) means for any given positive number \(\epsilon\), there is a number \(N\) such that for all \(n > N\), \(|a_n - \ell| < \epsilon\).
  • In sequence (b), the limit \(\ell = \frac{1}{2}\) means that the sequence terms \(\frac{n^3 + (-1)^n}{2n^3}\) approach \(\frac{1}{2}\) as \(n\) grows.
  • Visualizing this concept helps clarify why the sequences' behavior at larger \(n\) values demonstrate convergence.
Understanding the limit of a sequence helps in identifying the long-term trend of the sequence's behavior.
Sequence Simplification
Simplifying a sequence is crucial to identify its properties and analyze its behavior accurately. The process involves transforming the term expression into a more manageable form. Simplification often reveals how terms behave as \(n\) increases.

  • We simplify by dividing terms or canceling out terms to make the behavior of \(a_n\) clearer.
  • For instance, in sequence (a) \(a_n = \frac{n^2-1}{n^2+1}\), simplifying to \(1 - \frac{2}{n^2+1}\) quickly shows the approach to a limit of 1.
  • For sequence (b), converting \(a_n\) to \(\frac{1}{2} + \frac{(-1)^n}{2n^3}\) displays terms approaching \(\frac{1}{2}\), thus revealing its convergence characteristics.
Effective simplification can make convergence proofs and comprehension of the sequence far more accessible.

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Most popular questions from this chapter

For each of the following sequences \(\left\\{a_{n}\right\\}\), prove that \(a_{n} \rightarrow \infty:\) (a) \(\left\\{\frac{2^{n}}{n}\right\\}\); (b) \(\left\\{2^{n}-n^{100}\right\\}\) (c) \(\left\\{\frac{2^{2}}{n}+5 n^{100}\right\\}\); (d) \(\left\\{\frac{2^{n}+n^{2}}{n^{10}+n}\right\\} .\)

Show that each of the following sequences \(\left\\{a_{n}\right\\}\) is convergent, and find its limit: (a) \(a_{n}=\frac{n^{3}+2 n^{2}+3}{2 n^{3}+1}\); (b) \(a_{n}=\frac{n^{2}+2^{n}}{3^{n}+n^{3}}\); (c) \(a_{n}=\frac{n !+(-1)^{n}}{2^{n}+3 n !}\).

Prove that the following sequences are null: (a) \(\left\\{\frac{1}{n^{2}+n}\right\\}\) (b) \(\left\\{\frac{(-1)^{n}}{n !}\right\\}\); (c) \(\left\\{\frac{\sin n^{2}}{n^{2}+2^{n}}\right\\} .\)

(a) Calculate the first five terms of each of the following sequences: (i) \(\\{3 n+1\\}\) (ii) \(\left\\{3^{-n}\right\\}\); (iii) \(\left\\{(-1)^{n} n\right\\} .\) (b) Calculate the first five terms of each of the following sequences \(\left\\{a_{n}\right\\}:\) (i) \(a_{n}=n !, n=1,2, \ldots ;\) (ii) \(a_{n}=\left(1+\frac{1}{n}\right)^{n}, n=1,2, \ldots\) (to 2 decimal places).

Show that the following sequences \(\left\\{a_{n}\right\\}\) are monotonic: (a) \(a_{n}=n !, n=1,2, \ldots ;\) (b) \(a_{n}=2^{-n}, n=1,2, \ldots ;\) (c) \(a_{n}=n+\frac{1}{n}, n=1,2, \ldots\)
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