Problem 64 - 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 \(\epsilon\) $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.001 $$

Step-by-Step Solution

Answer

The limit is 0, and \( N = 32 \).

Step 1: Understand the Sequence

The sequence given is \( a_n = \frac{1}{n^2} \). As \( n \) increases, the value of \( a_n \) decreases because the denominator increases, making \( a_n \) approach zero.

Step 2: Determine the Limit

To find the limit of the sequence \( a_n \) as \( n \rightarrow \infty \), observe that as \( n \) grows very large, \( \frac{1}{n^2} \) approaches zero. Hence, the limit \( a \) is 0. Thus, \( \lim_{n \rightarrow \infty} a_n = 0 \).

Step 3: Set Up the Relationship

We need to find \( N \) such that \( |a_n - a| < \epsilon \) for all \( n > N \), where \( a = 0 \) and \( \epsilon = 0.001 \). This simplifies to \( \left| \frac{1}{n^2} - 0 \right| < 0.001 \).

Step 4: Solve for n

Rewrite the inequality: \( \frac{1}{n^2} < 0.001 \). This leads to the equation: \( n^2 > 1000 \).

Step 5: Calculate N

To find \( N \), solve \( n^2 > 1000 \) by taking the square root of both sides, yielding \( n > \sqrt{1000} \). Calculating \( \sqrt{1000} \) gives approximately 31.62. Thus, \( n \) must be greater than 31.62, so we choose the smallest integer greater than 31.62, which is 32. Therefore, \( N = 32 \).

Key Concepts

Convergence

In mathematics, the concept of convergence is essential when dealing with sequences and series. Convergence refers to the behavior of a sequence as its terms progress towards a specific value. This value is called the "limit". In simpler terms, when the terms of a sequence get closer and closer to a particular number, we say the sequence converges to that number.

For example, look at the sequence defined by \( a_n = \frac{1}{n^2} \). As \( n \), the term number, becomes larger and larger, the value of \( a_n \) becomes smaller and smaller, approaching zero. Therefore, we say the sequence \( a_n \) converges to 0. Here, 0 is the limit of the sequence. The convergence of this sequence is predictable and follows a pattern, which is typical in convergent sequences.

Why care about convergence? Knowing if a sequence converges helps in understanding the long-term behavior of sequences. This is crucial in analysis and helps ensure calculations involving infinite processes are meaningful.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define the limit of a sequence. It is a precise mathematical method to demonstrate that a sequence is converging to a limit. This concept is fundamental in explaining what it means for sequences to approach a limit accurately.

In the context of our sequence \( a_n = \frac{1}{n^2} \), the aim is to ensure the sequence terms come close to the limit, which is 0, as \( n \) becomes very large. According to the epsilon-delta definition, for a given small positive number \( \epsilon \), we need to find a value \( N \) such that for all \( n > N \), the inequality \(|a_n - a| < \epsilon\) holds.

For this exercise, with \( \epsilon = 0.001 \), the sequence satisfies the condition \(|\frac{1}{n^2} - 0| < 0.001\). This is achieved when \( n > 31.62 \), which we rounded up to the nearest whole number, resulting in \( N = 32 \). The epsilon-delta approach provides a way to measure how quickly a sequence approaches its limit.

Infinite Limit

An infinite limit describes the behavior of a sequence or function as the input grows indefinitely larger. It's a core concept in understanding how sequences behave at their extremes. However, it's important to distinguish it from the limit of a sequence as it approaches a finite number, often addressed with the epsilon-delta definition.

In the exercise example, the concept of an infinite limit is somewhat implicit, as we are considering the sequence \( a_n = \frac{1}{n^2} \) when \( n \rightarrow \infty \). The term "infinite" here captures the idea of considering the sequence's behavior as \( n \) grows without bound.

Infinite limits are vital in calculus to describe phenomena like asymptotic behavior, where sequences or functions approach a boundary but never quite reach it as they stretch into infinity. Understanding infinite limits is crucial for dealing with real-world scenarios, such as predicting behaviors over extended periods or understanding trends that stretch beyond measurable bounds.