Problem 56 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{n^{2}} $$

Step-by-Step Solution

Answer

The sequence converges to 1.

Step 1: Understand the Sequence Definition

The sequence given is \(a_n = \sqrt[n]{n^2}\). This means taking the \(n\)-th root of \(n^2\) for each term in the sequence.

Step 2: Consider the Behavior as n Approaches Infinity

To determine convergence or divergence, analyze the behavior of \(a_n\) as \(n\) approaches infinity. As \(n\) becomes very large, we suspect each term \(n^{2/n}\) contributes less to the growth of \(a_n\).

Step 3: Simplify the Expression for Large n

Rewriting \(a_n\), we have \(a_n = n^{2/n}\). Consider simplifying using the expression \(n^{1/n}\), which suggests evaluating \(n^{2/n} = (n^{1/n})^2\).

Step 4: Evaluate n to the Power of 1/n

For large values of \(n\), \(n^{1/n} \) tends towards \(1\). Thus, \((n^{1/n})^2\) also tends towards \(1^2 = 1\).

Step 5: Conclude on Convergence

Since \(n^{2/n} = (n^{1/n})^2\) approaches \(1\) as \(n\) approaches infinity, \(a_n\) converges to \(1\).

Step 6: Write the Conclusion

The sequence \(a_n = \sqrt[n]{n^2}\) converges, and the limit of the sequence is \(1\).

Key Concepts

Limit of a Sequence

The concept of the _limit of a sequence_ lies at the heart of understanding sequence convergence. A sequence is a list of numbers in a specific order, and the limit is the value the sequence approaches as the number of terms increases indefinitely. For example, in the sequence \( a_n = \sqrt[n]{n^2} \), we are interested in what happens to \( a_n \) as \( n \) becomes very large.

To find the limit, we often look at the behavior of the sequence's terms. In this case, we simplify \( a_n = n^{2/n} \) using the expression \( n^{1/n} \). As \( n \) increases, \( n^{1/n} \) tends towards \( 1 \). Therefore, \( (n^{1/n})^2 \), and consequently \( a_n \), approaches \( 1 \).

This process of analyzing each term's behavior helps us determine that \( a_n \) converges to \( 1 \) as its limit.

Infinite Sequences

Infinite sequences are sequences that continue indefinitely without ending. They consist of an ordered set of numbers, which can sometimes appear random or follow a calculable pattern.

In the problem \( a_n = \sqrt[n]{n^2} \), we see that as \( n \) gets infinitely large, the sequence aims to approach a specific value. The term "infinite" here signifies the limitless extent of the sequence, providing the basis for determining convergence or divergence.

Understanding infinite sequences is essential as it helps us predict the behavior of sequences in the limitless future, beyond just observing initial terms.

Convergence and Divergence

Convergence and divergence describe the behavior of infinite sequences. A sequence is said to converge if its terms approach a specific value (the limit) as they extend toward infinity. In contrast, a sequence diverges if its terms fail to settle towards any single value.

In our sequence \( a_n = \sqrt[n]{n^2} \), we determine convergence by simplifying \( a_n \) to see that it approaches \( 1 \) as \( n \) becomes very large. Thus, \( a_n \) is a convergent sequence with a limit of \( 1 \).

Diverging sequences, on the other hand, may increase, decrease, or oscillate endlessly without approaching a fixed value. Recognizing whether a sequence converges or diverges is crucial for deeper mathematical analysis.