Problem 63 - 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 \(\epsilon\) $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.01 $$

Step-by-Step Solution

Answer

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

Step 1: 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 \( a = 0 \).

Step 2: 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 \( \frac{1}{n^2} < 0.01 \).

Step 3: 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 \( n > 10 \).

Step 4: 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.

Key Concepts

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.