Problem 54 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: \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

The limit is 0, and \( N = 50 \) ensures \( \left|a_n - a\right| < 0.02 \) for all \( n > 50 \).

Step 1: Identify the Limit

The sequence provided is \( a_n = \frac{1}{n} \). As \( n \) approaches infinity, the term \( \frac{1}{n} \) approaches zero because the denominator grows without bound. Thus, the limit \( a \) is 0, i.e., \( \lim_{n \rightarrow \infty} a_n = 0 \).

Step 2: Apply the Formal Definition of Limit

According to the formal definition of a limit, for an \( \epsilon > 0 \), there exists an \( N \) such that \( |a_n - a| < \epsilon \) for all \( n > N \). In this exercise, \( a = 0 \) and \( a_n = \frac{1}{n} \), so the inequality becomes \( \left| \frac{1}{n} - 0 \right| < 0.02 \).

Step 3: Solve the Inequality for n

The inequality \( \left| \frac{1}{n} \right| < 0.02 \) simplifies to \( \frac{1}{n} < 0.02 \). To solve for \( n \), take the reciprocal: \( n > \frac{1}{0.02} = 50 \). Thus, for \( n > 50 \), \( \left| \frac{1}{n} \right| < 0.02 \) holds true.

Step 4: Determine the Value of N

From the previous step, we have found that \( n > 50 \). Therefore, we can set \( N = 50 \). For all \( n > N \), the condition \( \left| \frac{1}{n} \right| < \epsilon \) is satisfied, where \( \epsilon = 0.02 \).

Key Concepts

Formal Definition of Limits

When we talk about limits, we're essentially discussing the behavior of a function or sequence as it approaches a particular point. The formal definition of a limit is a rigorous way to capture this notion. It tells us that if you have a sequence like \( a_n \), as \( n \) becomes very large (approaches infinity), the sequence's terms get closer to a specific value, called the limit. In the formal definition, for every small positive number \( \epsilon \), there exists a corresponding integer \( N \) such that for all \( n > N \), the distance or difference between \( a_n \) and its limit \( a \) is less than \( \epsilon \). Mathematically, it is represented as:

• \( |a_n - a| < \epsilon \) for all \( n > N \).

Here, \( \epsilon \) represents how close you want \( a_n \) to be to \( a \). This definition of a limit ensures that the sequence doesn't just get close to \( a \) once; it stays close beyond that point.

Sequence

A sequence is a list of numbers written in a specific order, following a particular pattern or rule. For example, in the original exercise, the sequence given was \( a_n = \frac{1}{n} \). As \( n \) increases, each term \( a_n \) decreases, getting closer to 0.Sequences can have various properties:

• Convergent Sequence: A sequence that approaches a specific value as \( n \) gets large is known as convergent.
• Divergent Sequence: A sequence that does not approach any particular value is called divergent.

In this exercise, the sequence \( \frac{1}{n} \) is a classic example of a convergent sequence, as it approaches 0. Understanding sequences is key, as they help us interpret and apply the concept of limits effectively.

Epsilon

The concept of \( \epsilon \) is crucial in the formal definition of limits. It represents an arbitrary positive distance from the limit \( a \), allowing us to quantify how close the sequence \( a_n \) must get to \( a \) from some point onward.In our specific example, we have \( \epsilon = 0.02 \), meaning:

• The terms of the sequence \( a_n = \frac{1}{n} \) must be within 0.02 units of the limit, which is 0.
• This condition translates into the inequality \( \left| \frac{1}{n} \right| < 0.02 \).

The role of \( \epsilon \) can be thought of as the tolerance level for the sequence's closeness to the limit. By choosing different values for \( \epsilon \), you can test the robustness of the sequence's convergence. It's a way to gain more control and precision in approaching limits.