Problem 55 - 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$. $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.01 $$

Step-by-Step Solution

Answer

The limit is 0, and \(N = 10\).

Step 1: Analyze Limit

We begin by finding the limit of the sequence as \( n \to \infty \) when \( a_n = \frac{1}{n^2} \). As \( n \) becomes very large, \( \frac{1}{n^2} \) approaches 0 because the denominator increases much faster than the numerator. Thus, \( \lim_{n \to \infty} a_n = 0 \), so \( a = 0 \).

Step 2: Set Up Inequality

Next, we use the definition of the limit to set up an inequality with our calculated limit \( a = 0 \) and given \( \epsilon = 0.01 \). We need \( |a_n - a| < \epsilon \), which becomes \( \left| \frac{1}{n^2} - 0 \right| < 0.01 \). This simplifies to \( \frac{1}{n^2} < 0.01 \).

Step 3: Solve Inequality for n

To solve \( \frac{1}{n^2} < 0.01 \), first invert the inequality as \( n^2 > \frac{1}{0.01} \). Simplifying, we get \( n^2 > 100 \). Taking the square root of both sides yields \( n > 10 \). Thus, \( N = 10 \) serves as our threshold value.

Key Concepts

Formal Definition of Limits

In calculus, understanding the formal definition of limits is crucial for grasping the behavior of sequences and functions as inputs approach a particular value. To say that the limit of a sequence \( \lim_{n \rightarrow \infty} a_n = a \) implies that as \( n \) becomes very large, the terms of the sequence \( a_n \) get closer and closer to the number \( a \). Consider it as predicting where a sequence will "settle down" as it keeps going. This concept helps in finding where sequences converge and is fundamental to calculus.

• Intuition: Think of limits as a destination that the sequence terms are trying to reach as the index \( n \) grows infinitely large.
• Example: For the sequence \( a_n = \frac{1}{n^2} \), the terms become smaller and smaller, approaching 0 as \( n \) increases.

This understanding of limits is not just an abstract mathematical concept; it is applied extensively in calculus to study continuity, derivatives, and integrals.

Sequence Convergence

Sequence convergence refers to the behavior of a sequence as its terms increasingly approach a specific value, known as the limit. A sequence \( \{a_n\} \) is said to converge to \( a \) if, for every positive number \( \epsilon \), there exists a point in the sequence beyond which all terms are within \( \epsilon \) of \( a \). In simpler terms, that's like saying, after a certain point, all terms of the sequence are close enough to the limit.

• Convergent vs. Divergent: While convergent sequences approach a specific value, divergent sequences do not—they might oscillate or grow without bound.
• Example: The sequence \( a_n = \frac{1}{n^2} \) approaches 0 as \( n \to \infty \), and this is verified by observing that the fraction's value gets increasingly small.

Understanding sequence convergence helps predict long-term behavior, and it's a foundational skill in analyzing series and functions in calculus.

Epsilon-Delta Definition

The epsilon-delta definition is a rigorous method used to define limits in mathematics. It provides a precise way to show that a function or sequence converges to a limit. The definition states that \( \lim_{n \to \infty} a_n = a \) if for every positive number \( \epsilon \), there exists a corresponding \( N \) such that for all \( n > N \), the absolute difference \( |a_n - a| < \epsilon \). This directly ties into proving sequence convergence.

• Practical Approach: To apply this definition, one must find an appropriate \( N \) that satisfies the condition for any given \( \epsilon \).
• Example: In our exercise, for \( a_n = \frac{1}{n^2} \) and \( \epsilon = 0.01 \), solving \( |\frac{1}{n^2} - 0| < 0.01 \) gives \( n > 10 \) as the smallest integer \( N \).

Mastering the epsilon-delta definition provides a comprehensive understanding of how limits work in mathematical analysis, ensuring precise calculations and conclusions about sequences and continuous functions.