Problem 21 - Precalculus: Mathematical for Calculus

Question

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n^{2}}{n+1}$$

Step-by-Step Solution

Answer

The sequence is divergent because it approaches infinity as \(n\) increases.

Step 1: Analyze the Sequence

We need to determine if the sequence \(a_n = \frac{n^2}{n+1}\) converges. To do this, we'll evaluate the behavior of the sequence as \(n\) approaches infinity.

Step 2: Simplify the Fraction

Divide the numerator and the denominator by \(n\) to simplify the sequence:\[a_n = \frac{n^2}{n+1} = \frac{n}{1 + \frac{1}{n}}.\]As \(n\) grows large, the term \(\frac{1}{n}\) approaches 0.

Step 3: Evaluate the Limit as \(n\) Tends to Infinity

Take the limit of \(a_n\) as \(n\) approaches infinity:\[\lim_{{n \to \infty}} \frac{n}{1 + \frac{1}{n}} = \lim_{{n \to \infty}} \frac{n}{1} = n.\]This shows that the sequence diverges to infinity.

Step 4: Conclusion

Since the limit of the sequence \(a_n\) as \(n\) approaches infinity is infinite, the sequence is divergent.

Key Concepts

Understanding the Limit of a Sequence

The concept of the limit of a sequence is essential to understanding how sequences behave as they progress towards infinity. To determine whether a sequence converges or diverges, we examine its behavior as the values approach a certain number, possibly infinity.
A sequence is said to converge to a limit if, as you move further along the sequence (by increasing the index, usually denoted by \( n \)), the sequence approaches a particular value. This value is the limit.
If a sequence does not approach any finite value, or it grows infinitely large, it is considered divergent.
In the given exercise, the sequence \( a_n = \frac{n^2}{n+1} \) was analyzed to see whether it converges to a finite number or diverges. By simplifying the expression as \( n \to \infty \), it was found that the sequence diverges due to growing infinitely large.

Exploring Infinite Limits

Infinite limits occur when the terms of a sequence increase without bound as \( n \) approaches infinity. This results in the sequence not approaching a finite value, hence it diverges.
In our exercise, the aim was to discover the behavior of \( a_n = \frac{n^2}{n+1} \) as \( n \to \infty \).
By simplifying the sequence, we evaluated the limit of \( \frac{n}{1} = n \) as \( n \to \infty \).

• We first divided the expression to handle the infinity terms correctly.
• The term \( \frac{1}{n} \) becomes negligible as \( n \) becomes very large.
• The limit \( n \to \infty \) simplifies to being infinite.

Such infinite behavior of sequences is a clear sign of divergence, illustrating an infinite limit.

Behavior of Sequences and Divergence

The behavior of sequences is a pivotal concept in determining their eventual outcomes, whether they converge or diverge. Understanding the progression of sequences helps tackle various mathematical problems and predict patterns.
Sequences can exhibit different types of behavior based on their formulas and limitations.

• Convergent sequences approach a certain finite number.
• Divergent sequences do not settle down to a single limit and can either oscillate without settling or grow indefinitely like in our example.

In the example \( a_n = \frac{n^2}{n+1} \), observing how terms behave as \( n \to \infty \) provided insights into the ultimate behavior of the sequence. Analyzing this helped us conclude that as the sequence grows, its behavior manifests in divergence, confirming that it does not converge to a finite limit.