Problem 75 - Calculus for Biology and Medicine

Question

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$

Step-by-Step Solution

Answer

For every \( \epsilon > 0 \), choose \( N = \left\lceil \frac{1}{\sqrt{\epsilon}} \right\rceil \) to satisfy the limit definition.

Step 1: Understanding the Limit Expression

We need to prove that \( \lim_{n \to \infty} \frac{1}{n^2} = 0 \) using the formal definition of a limit. The formal definition states that for every \( \epsilon > 0 \), there exists a number \( N \) such that if \( n > N \), then \( \left| \frac{1}{n^2} - 0 \right| < \epsilon \).

Step 2: Simplify the Expression

First, notice that \( \left| \frac{1}{n^2} - 0 \right| = \left| \frac{1}{n^2} \right| = \frac{1}{n^2} \). This simplifies our task to finding \( N \) such that \( \frac{1}{n^2} < \epsilon \) for all \( n > N \).

Step 3: Find the Inequality Condition

We want \( \frac{1}{n^2} < \epsilon \). To solve for \( n \), take the reciprocal of both sides (remembering to reverse the inequality): \( n^2 > \frac{1}{\epsilon} \).

Step 4: Determine the Value of N

Taking the square root of both sides gives \( n > \frac{1}{\sqrt{\epsilon}} \). Thus, we can choose \( N = \left\lceil \frac{1}{\sqrt{\epsilon}} \right\rceil \), where \( \lceil x \rceil \) represents the smallest integer greater than or equal to \( x \). This ensures that \( n > N \) will satisfy our condition.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta (\(\epsilon\)-\(\delta\)) definition is an essential concept in calculus, particularly when discussing limits. This formal definition provides a way to rigorously prove that a function approaches a specific value as its input approaches some value. For sequences, the definition is slightly adapted. Here’s how it works:

• Given a sequence \(a_n\) that converges to \(a\), the goal is to show that for every arbitrarily small positive number \(\epsilon\), we can find an integer \(N\) such that all terms of the sequence beyond the \(N\)-th term are within \(\epsilon\) of \(a\).


• In simpler terms, we want \(\left|a_n - a\right| < \epsilon\) whenever \(n > N\).

With this definition, if we argue that these conditions are met for every \(\epsilon\), then \(a_n\) converges to \(a\).In our original exercise, we applied this definition to prove that the limit of \(\frac{1}{n^2}\) as \(n\) approaches infinity is indeed 0.

Convergence in Sequences

Convergence in sequences describes how a sequence of numbers approaches a particular value as the number of terms increases. To say that a sequence converges, every term beyond a certain point should approximate the limit more closely.

• A sequence \(a_n\) converges to a limit \(a\) if the terms get arbitrarily close to \(a\) as \(n\) becomes very large.


• The epsilon-delta definition provides the framework to prove this convergence rigorously.

In our problem, the sequence \(\frac{1}{n^2}\) converges to 0 as \(n\) approaches infinity. Using the epsilon-delta definition, we demonstrated that for any small number \(\epsilon\), you can always find an integer \(N\) such that \(\frac{1}{n^2}\) gets closer than \(\epsilon\) to 0 for all \(n > N\). Understanding this process helps in mastering the idea of limits and how sequences behave as they approach their respective limits.

Limits at Infinity

When discussing limits at infinity, we look at the behavior of a sequence or function as its input grows infinitely large. This is crucial in understanding how sequences settle into their limiting value over time.

• The goal is to determine the value that a sequence approaches as \(n\) becomes very large.


• For the sequence \(\frac{1}{n^2}\), the exercise illustrated that as \(n\) increases infinitely, \(\frac{1}{n^2}\) approaches 0.

This concept helps understand real-world phenomena where values become negligible or settle around a stable point as time progresses or sizes grow larger. The step-by-step solution explained how to choose an appropriate \(N\) so that the sequence lies within a desired range of its limit for any small \(\epsilon\). This showcases not only the mathematics involved but also how limits ensure predictability in infinite processes.