Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


e.The sequence aₙ = n² / (n² + 1) converge.

Verified step by step guidance

1. Identify the sequence given: \(a_n = \frac{n^2}{n^2 + 1}\).
2. Recall the definition of convergence for a sequence: a sequence \(a_n\) converges to a limit \(L\) if \(\lim_{n \to \infty} a_n = L\) exists and is finite.
3. To determine if \(a_n\) converges, compute the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{n^2}{n^2 + 1}\).
4. Divide numerator and denominator by \(n^2\) to simplify the expression: \(\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n^2}}\).
5. Evaluate the limit by noting that \(\frac{1}{n^2} \to 0\) as \(n \to \infty\), so the limit becomes \(\frac{1}{1 + 0} = 1\), which means the sequence converges to 1.

Key Concepts

Sequence Convergence

A sequence converges if its terms approach a specific finite value as the index n approaches infinity. This means for any small positive number, the terms eventually get arbitrarily close to that value. Understanding convergence helps determine the long-term behavior of sequences.

Limit of a Sequence

The limit of a sequence is the value that the terms approach as n becomes very large. Calculating the limit often involves simplifying the expression and analyzing dominant terms. If the limit exists and is finite, the sequence converges to that limit.

Dominant Term Analysis

When evaluating limits of sequences involving polynomials, the highest degree terms dominate the behavior as n grows large. By comparing the degrees of numerator and denominator, one can simplify the expression to find the limit, which is crucial for determining convergence.