Problem 53 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: In Problems \(53-64, \lim _{n \rightarrow \infty} a_{n}=a .\) Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\). $$ a_{n}=\frac{1}{n}, \epsilon=0.01 $$

Step-by-Step Solution

Answer

The limit is 0, and \( N = 100 \) ensures \( |a_n - 0| < 0.01 \) for all \( n > 100 \).

Step 1 - Recognize the Formula

We start by recognizing the sequence we are dealing with, which is given as \( a_n = \frac{1}{n} \). Our goal is to find the limit \( a \) as \( n \to \infty \) and determine \( N \) such that \( |a_n - a| < \epsilon \) for all \( n > N \).

Step 2 - Find the Limit

As \( n \to \infty \), the term \( \frac{1}{n} \) approaches 0 because the numerator stays constant while the denominator becomes infinitely large. Therefore, the limit \( a \) is 0. So, \( \lim_{n \to \infty} a_n = 0 \).

Step 3 - Set Up the Inequality

We need to satisfy the condition \( \left| \frac{1}{n} - 0 \right| < \epsilon \). Simplifying, this becomes \( \frac{1}{n} < 0.01 \).

Step 4 - Solve for N

To find \( N \), solve the inequality \( \frac{1}{n} < 0.01 \). This means \( n > \frac{1}{0.01} = 100 \). Therefore, \( N = 100 \).

Step 5 - Verify the Solution

Let's verify that if \( n = 101 \), then \( \frac{1}{n} = \frac{1}{101} \approx 0.0099 \), which is indeed less than 0.01. Hence, for all \( n > 100 \), \( \frac{1}{n} < 0.01 \) confirming that our choice of \( N \) is correct.

Key Concepts

Formal Definition of a Limit

In calculus, understanding the formal definition of a limit is crucial as it sets the foundation for more advanced topics. The limit of a sequence or function describes what value it approaches as the input grows without bound. This concept can sometimes be hard to grasp due to its abstract nature, but it is essentially a way to predict behavior of a mathematical function without reaching the point directly.

When we say that the limit of a sequence \( a_n \) as \( n \to \infty \) is \( a \), we mean that as \( n \) increases towards infinity, the sequence \( a_n \) gets arbitrarily close to \( a \). For example, in the given sequence \( a_n = \frac{1}{n} \), as \( n \) becomes very large, \( \frac{1}{n} \) gets closer to 0. Thus, the limit \( a \) is 0. This method of identifying the behavior of sequences is a precise and powerful tool in calculus.

Sequence Convergence

Sequence convergence is all about whether a sequence approaches a specific value. When a sequence converges, its terms come closer and closer to a certain value, called the limit.

To determine convergence:

• Identify if the terms of the sequence are getting closer to a particular value.
• Check if the "distance" between the sequence terms and this value is shrinking as the sequence progresses.

Considering the sequence \( a_n = \frac{1}{n} \), it converges to 0 because the terms become smaller with increasing \( n \). For practical application, we can pinpoint a moment from a given point (like after 100 terms for this sequence) where all subsequent terms stay very close to this limit value.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way of thinking about limits and is vital for validating that a sequence truly reaches a limit. It offers a rigorous proof of convergence.
To use this definition:

• Begin with an arbitrary small positive number, \( \epsilon \), signifying how close you want the sequence terms to be to the limit.
• Find a corresponding \( N \), where for all \( n > N \), the terms \( |a_n - a| < \epsilon \).

This definition is applied to ensure precision. In the problem given, we need \( |\frac{1}{n} - 0| < 0.01 \), which implies \( \frac{1}{n} < 0.01 \) ensuring \( n > 100 \). Therefore, choosing \( N = 100 \) assures that for \( n > 100 \), the terms of the sequence are closer than 0.01 to the limit 0, confirming the precise convergence of the sequence.