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

Step-by-Step Solution

Answer

The limit \( a = 1 \) and \( N = 10 \).

Step 1: Understand the Expression

Given the sequence \( a_n = \frac{n^2}{n^2 + 1} \), we want to find its limit as \( n \to \infty \). This involves understanding the behavior of \( a_n \) for very large \( n \).

Step 2: Find the Limit of the Sequence

For large \( n \), the leading terms dominate, thus \( a_n \approx \frac{n^2}{n^2} = 1 \). Therefore, the limit \( a \) of the sequence is 1. So, \( \lim_{n \to \infty} a_n = 1 \).

Step 3: Define Close-enough Criterion

We are given that \( \epsilon = 0.01 \), meaning we want \( |a_n - a| = |a_n - 1| < 0.01 \) for all \( n > N \).

Step 4: Solve the Inequality for |a_n - a|

Start with the equation \( |\frac{n^2}{n^2 + 1} - 1| < 0.01 \).Rewriting gives: \( \left| \frac{n^2 - (n^2 + 1)}{n^2 + 1} \right| < 0.01 \), or \( \left| \frac{-1}{n^2 + 1} \right| < 0.01 \). This simplifies to \( \frac{1}{n^2 + 1} < 0.01 \).

Step 5: Simplify and Solve for N

From \( \frac{1}{n^2 + 1} < 0.01 \), multiply both sides by \( n^2 + 1 \) and then by 100 to isolate \( n^2 + 1 \):\( 1 < 0.01 (n^2 + 1) \) \ \( 100 < n^2 + 1 \) \ \( n^2 > 99 \) \ \( n > \sqrt{99} \).

Step 6: Calculate N

By taking the square root: \( n > \sqrt{99} \approx 9.95 \).Since \( n \) must be an integer, we take \( n > 10 \), so choose \( N = 10 \). Thus, for \( n > 10 \), \( |a_n - 1| < 0.01 \).

Key Concepts

Formal Definition of Limits

In calculus, the formal definition of limits is fundamental for understanding how a function behaves as it approaches a specific point. The limit of a sequence \( \lim_{n \to \infty} a_{n} = a \)describes the value that the terms of a sequence, \( a_n \), approach as \( n \) goes to infinity.

To apply this definition, one must identify a point, \( a \), that is the target value the sequence is approaching. Understanding and proving this involves logical and often numerical reasoning. It requires us to demonstrate that as \( n \) increases indefinitely, the sequence values become arbitrarily close to \( a \). This enables the evaluation not just of practical calculations but also of more complex mathematical behaviors.

Epsilon-Delta Definition

A core concept in calculus, the epsilon-delta definition provides a rigorous way of defining limits. It offers a precise mathematical framework to prove that the limit of a sequence, \( a_n \), truly approaches a specific value, \( a \).

In this definition, given any small positive number \( \epsilon \) (epsilon), there exists a corresponding number \( N \) such that for all \( n > N \),

• \( \left| a_n - a \right| < \epsilon \)

This inequality states that the distance between the terms of the sequence and the limit \( a \) is less than \( \epsilon \), making them extremely close.
The epsilon-delta definition is instrumental for proving the convergence of sequences and ensures that sequences can be characterized with mathematical precision.

Sequence Behavior Analysis

Analyzing the behavior of a sequence involves observing how the terms behave as the sequence progresses, particularly as the index \( n \) increases. In many cases, like with the sequence \( a_n = \frac{n^2}{n^2 + 1} \),one must simplify the expression to find the limit behavior for large values of \( n \).

When approaching this sequence's characteristics, we notice that as \( n \) becomes very large, the dominant term in the polynomial, \( n^2 \), overshadows the other components.

• This results in \( a_n \approx 1 \), indicating that the sequence terms are approaching a value of 1.

Understanding this simplification by identifying the influence of prominent terms in the equation is crucial to correctly predict how the sequence behaves, especially as it approaches infinity.

Limit of a Sequence

Finding the limit of a sequence involves determining the value that the terms in the sequence tend towards as the number of terms, \( n \), goes to infinity.

For the sequence \( a_n = \frac{n^2}{n^2 + 1} \), as \( n \to \infty \), the limit is determined by examining how the sequence's terms evolve:

• Given the expression, the fraction simplifies to \( 1 \) as \( n \) grows larger since \( n^2 \) becomes much larger than 1, making the sequence's behavior dominated by \( n^2/n^2 \).
• The limit here is thus \( 1 \), confirmed by logical simplification and analysis.

Calculating the limit of a sequence is essential for understanding its long-term trend and convergence, and is applicable in various areas of mathematics and its applications.