Problem 62 - Calculus for Biology and Medicine

Question

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{n}, \epsilon=0.02 $$

Step-by-Step Solution

Answer

Limit is 0, and \(N = 50\).

Step 1: Identify the Sequence Limit

Given the sequence, \( a_n = \frac{1}{n} \), observe that as \( n \to \infty \), \( \frac{1}{n} \to 0 \). Hence, the limit \( a \) is \( 0 \).

Step 2: Establish the Inequality

We are given \( \epsilon = 0.02 \). We need to find \( N \) such that for all \( n > N \), \( \left| a_n - a \right| < \epsilon \). This means \( \left| \frac{1}{n} - 0 \right| < 0.02 \), which simplifies to \( \frac{1}{n} < 0.02 \).

Step 3: Solve the Inequality for n

Rearrange the inequality \( \frac{1}{n} < 0.02 \) to find the threshold for \( n \). This gives \( n > \frac{1}{0.02} \), which simplifies to \( n > 50 \).

Step 4: Determine N

From the inequality \( n > 50 \), we conclude that \( N = 50 \). For all \( n > 50 \), the inequality \( \left| \frac{1}{n} - 0 \right| < 0.02 \) will hold.

Key Concepts

Convergence of Sequences

The idea of convergence in sequences is fundamental in calculus. A sequence is essentially an ordered list of numbers, and convergence refers to the behavior of this list as it extends towards infinity. When we say a sequence converges, we mean that as we continue along the list, the numbers get closer and closer to a specific number, called the limit.
For example, the sequence given in the exercise is \( a_n = \frac{1}{n} \). As \( n \), or the position in our list, becomes very large, \( \frac{1}{n} \) approaches 0. This means the sequence converges to 0.
In the convergence of sequences, we analyze whether the numbers in the sequence approach some particular value as the number of terms increases without bound. Understanding convergence is crucial as it establishes the groundwork for further exploration in calculus, such as series analysis and continuous functions.

Epsilon-Delta Definition

The epsilon-delta definition is a precise mathematical formulation used to describe the limit of a function or sequence. Though often thought of in the context of function limits, it can be equally applied to sequences. In our exercise, we needed to find \( N \) such that for all \( n > N \), \( |a_n - a| < \epsilon \). Here, \( \epsilon \) is a small positive number representing the tolerance level of closeness between the sequence \( a_n \) and its limit \( a \).
Understanding the epsilon-delta definition might seem daunting at first, but it essentially ensures that the sequence gets increasingly close to the limit and stays there. Break it down into simple parts:

• \( \epsilon \): A small number indicating how close we want the sequence terms to be to the limit.
• \( |a_n - a| \): The absolute difference between the terms of the sequence and the limit.
• \( N \): A threshold after which all terms of the sequence are within \( \epsilon \) of the limit.

For the sequence \( a_n = \frac{1}{n} \) and \( \epsilon = 0.02 \), we found that \( N = 50 \). This means starting from \( n = 51 \), every term of the sequence will be within 0.02 of 0.

Sequence Analysis

Sequence analysis involves understanding the properties and behaviors of sequences. This analysis helps in comprehending broader mathematical concepts and plays a crucial role in calculus. In our exercise, we analyzed the sequence \( a_n = \frac{1}{n} \) for its limit and the behavior as \( n \) becomes large.
Analyzing sequences involves several steps:

• Identifying a pattern in the sequence, such as the formula \( a_n = \frac{1}{n} \).
• Determining the limit \( a \). For \( \frac{1}{n} \), this limit is \( 0 \) as \( n \to \infty \).
• Applying conditions like the epsilon-delta criterion to establish values like \( N \), ensuring all terms beyond \( N \) are within acceptable distance from the limit.

This approach teaches students not only to find the limit but also to understand when a sequence behaves consistently within specified boundaries. Through sequence analysis, one can predict and confirm the expected behavior of sequences and explore their convergence, a process integral to mastering calculus. This understanding becomes especially useful when addressing more complex sequences and functions.