Problem 6 - Basic Analysis: Introduction to Real Analysis

Question

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{n}{n^{2}+1}\right\\}\) convergent? If so, what is the limit?

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Identify the sequence and its behavior

Given the sequence \(a_n = \frac{n}{n^2 + 1}\) we need to determine whether it converges or diverges. The key to understanding this is to analyze the behavior of the sequence as \(n\) approaches infinity.

Step 2: Examine the sequence at infinity

As \(n\) grows very large, the sequence \(a_n = \frac{n}{n^2 + 1}\) simplifies to \(a_n \approx \frac{n}{n^2} = \frac{1}{n}\). This approximation is because the \(n^2\) term in the denominator dominates, making \(n^2 + 1 \approx n^2\).

Step 3: Determine the limit of the sequence

Given the approximation \(a_n \approx \frac{1}{n}\), we can find the limit as \(n\) approaches infinity. Since \(\frac{1}{n} \to 0\) as \(n\to\infty\), this suggests that \(\lim_{{n \to \infty}} a_n = 0\).

Step 4: Prove the correctness of the limit

To rigorously prove this limit, for any \(\varepsilon > 0\), we need to find \(N\) such that \(n > N\) implies \(|a_n - 0| < \varepsilon\). Observe that \( |a_n - 0| = \frac{n}{n^2 + 1} < \frac{n}{n^2} = \frac{1}{n}\). Choose \(N > \frac{1}{\varepsilon}\), then for any \(n > N\), \( \frac{1}{n} < \varepsilon\). This proves \(\lim_{{n \to \infty}} \frac{n}{n^2 + 1} = 0\).

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus used to understand the behavior of functions as they approach a specific point or value—often infinity. When dealing with sequences, limits help us predict the long-term behavior as the sequence progresses.
In the given exercise, finding the limit of the sequence \(a_n = \frac{n}{n^2 + 1}\) involves analyzing what happens to \(a_n\) as \(n\) becomes very large. By simplifying the expression for large \(n\), it was observed that \(a_n\) approximates to \(\frac{1}{n}\).
This approximation is crucial because the limit of \(\frac{1}{n}\) as \(n\) grows without bound is 0. Thus, using this knowledge from calculus, we rigorously concluded that \(\lim_{{n \to \infty}} a_n = 0\).
Calculus tools like L'Hôpital's rule, derivatives, and continuity often extend to analyzing limits, but for sequences, especially rational ones like in this exercise, simplification and comparison to known limits are key strategies.

Behavior of Sequences

Understanding the behavior of sequences involves predicting their patterns and determining whether they stabilize or continue to fluctuate without bounds. Behavior analysis can reveal much about a sequence, such as whether it converges (approaches a limit) or diverges (fails to approach any limit).
For the sequence \(a_n = \frac{n}{n^2 + 1}\), analyzing its behavior at infinity was essential. It led us to see how the denominator grows much faster than the numerator, implying a diminishing effect over time. We used the simplification \(\frac{n}{n^2} = \frac{1}{n}\) to reveal the sequence's convergent nature.
Detecting this behavior is useful, not just to prove convergence but to understand how quickly the sequence approaches its limit. Often, additional terms or factors could change this behavior slightly, but the trend remains governed by the dominant terms in numerator and denominator as \(n\) becomes large.

Infinite Sequences

Infinite sequences are ordered lists of numbers that extend indefinitely. When analyzing these sequences, our primary goal is to determine if they have a limit, which would imply that the sequence converges to a specific number.
In this exercise, the infinite sequence \(\left\{\frac{n}{n^2+1}\right\}\) was under study. Through careful examination, even at infinite lengths, we showed that the sequence becomes arbitrarily close to the number 0.
Since infinity doesn’t provide a stopping point, mathematicians consider what value the sequence appears to settle into or near. Tools like convergence tests and limit laws ensure we rigorously conclude whether a limit exists.
Infinite sequences are common in mathematics as they underpin series, analysis, number theory, and practical applications like computer science algorithms where they're often used to model processes or behaviors that repeat endlessly.