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Chapter 2: Problem 63

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.01 $$

Short Answer

Expert verified
The limit is 0, and for all \( n > 10 \), \( \left| a_n - a \right| < 0.01 \) holds.

Step by step solution

01

Identify the Limit

We begin by observing the sequence \( a_n = \frac{1}{n^2} \). As \( n \) approaches infinity, \( \frac{1}{n^2} \) approaches 0 because the denominator \(n^2\) grows unbounded. Therefore, the limit is \( a = 0 \).
02

Set Up the Inequality

Our goal is to find \( N \) such that for all \( n > N \), \( \left| a_n - a \right| < \epsilon \). This means we want \( \left| \frac{1}{n^2} - 0 \right| < 0.01 \), which simplifies to \( \frac{1}{n^2} < 0.01 \).
03

Solve for \( n \)

To solve \( \frac{1}{n^2} < 0.01 \), we first take the reciprocal of both sides, resulting in \( n^2 > 100 \). Taking the square root of both sides, we have \( n > 10 \).
04

Determine \( N \)

Since \( n > 10 \) ensures that our inequality holds, we can choose \( N = 10 \). Therefore, for all \( n > 10, \left| a_n - a \right| < 0.01 \) holds true.

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Key Concepts

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

Inequality
Inequalities are mathematical expressions that show the relationship of one value being larger or smaller than another value. In our context of sequences, inequalities help us determine how close or far away the terms of a sequence are from the limit.

For the sequence given by the formula \( a_n = \frac{1}{n^2} \), we use an inequality to establish when the differences \( |a_n - a| \) fall below a specified threshold \( \epsilon \).
  • The inequality \( |\frac{1}{n^2} - 0| < 0.01 \) translates to all \( n > N \) satisfying \( a_n \) being within \( 0.01 \) of the limit \( a \).
  • By simplifying, we find \( \frac{1}{n^2} < 0.01 \), which upon solving gives \( n^2 > 100 \) and thus, \( n > 10 \).

This solution outlines how we can confidently choose an appropriate \( N = 10 \) to ensure our sequence remains within this bound for all \( n > 10 \). By understanding inequalities, students can grasp how sequences "settle" into their limits.
Convergence
Convergence refers to the behavior of a sequence as its terms progressively approach a specific value, known as the limit, as the index increases indefinitely. A sequence is said to converge if the terms get arbitrarily close to a number, no matter how small the proximity required.

In our exercise, the sequence \( a_n = \frac{1}{n^2} \) is analyzed for convergence. The sequence approaches zero as \( n \) increases because:
  • The term \( \frac{1}{n^2} \) becomes smaller and smaller as \( n \) grows, since \( n^2 \) grows faster and approaches infinity.
  • This implies that as more terms are computed from this sequence, they will get closer to 0, which serves as the limit.
  • Thus, \( a_n \to 0 \) as \( n \to \infty \).
Understanding convergence helps students clearly see how a sequence can be controlled by its limit behavior, providing a powerful tool in analysis.
Epsilon-Delta Definition
The Epsilon-Delta Definition is the rigorous mathematical framework used to define limits and establish whether sequences or functions approach a certain limit. It formalizes the concept of convergence by providing a precise way to talk about how close terms of a sequence must be to the actual limit.

In simpler terms, the definition asserts:
  • For any small positive number \( \epsilon \) (no matter how small), there exists a corresponding large number \( N \) such that for all \( n > N \), the sequence terms \( |a_n - a| < \epsilon \).
  • For the sequence \( a_n = \frac{1}{n^2} \), given an \( \epsilon = 0.01 \), we've found that if \( n > 10 \), then \( |a_n - 0| < 0.01 \).

This framework provides the exact point \( N = 10 \) after which all sequence terms are confined within an \( \epsilon \)-neighborhood around the limit, showing precise convergence. It offers students a means to grasp how sequences change and adapt over increasing indices.

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Most popular questions from this chapter

You do not know whether a drug has zeroth order or first order elimination kinetics. You will use data to determine which type of kinetics it has. You measure the concentration of the drug (in \(\mathrm{mg} / \mathrm{ml}\) ) at time \(t=0\) and at time \(t=1 .\) No drug is added to the blood between \(t=0\) and \(t=1\). You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 40 \\ 1 & 32 \\ \hline \end{tabular} (a) Assume that the drug has zeroth order kinetics. What amount is eliminated from the blood each hour? (b) Assume that the drug has zeroth order kinetics and no more drug is added to the blood. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (c) Now assume the drug has first order elimination kinetics. What percentage of drug is eliminated from the blood each hour? (d) Assume that the drug has first order kinetics and no more drug is added. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 25.6. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(3^{-n}-4^{-n}\right) $$

A population obeys the Beverton-Holt model. You know that \(R_{0}=3\) for this population. As \(t \rightarrow \infty\) you observe that \(N_{t} \rightarrow 100 .\) What value of \(a\) is needed in the model to fit it to these data?

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(20 \cdot(0.2)^{t}\). (a) The drug has first order elimination kinetics. \(40 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show that, when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\frac{a_{n}}{1+a_{n}}, a_{0}=1 $$
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