Problem 65 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\sqrt[n]{4^{n} n} $$

Step-by-Step Solution

Answer

The sequence converges to 4.

Step 1: Rewrite the Sequence

The given sequence is \(a_n = \sqrt[n]{4^n n}\). We can rewrite this using exponent rules as follows: \(a_n = (4^n n)^{1/n} = 4 (n^{1/n})^{1/n}\).

Step 2: Analyze the Term \(n^{1/n}\)

Examine the behavior of \(n^{1/n}\) as \(n\) approaches infinity. As \(n\) becomes very large, \(n^{1/n}\) approaches 1 because the nth root of \(n\) (which is \(n^{1/n}\)) becomes very small.

Step 3: Evaluate Limit of the Entire Expression

Since \((n^{1/n})^{1/n} = n^{1/n^2} \rightarrow 1\) as \(n \to \infty\), then, \(a_n = 4 \cdot n^{1/n^2}\). Therefore, \(a_n \rightarrow 4 \cdot 1 = 4\) as \(n \to \infty\).

Step 4: Conclusion

As the sequence \(a_n = \sqrt[n]{4^n n} = 4 \cdot n^{1/n^2}\) converges to 4, we conclude that the sequence converges. The limit of the sequence is 4.

Key Concepts

Limit of a Sequence

Understanding the limit of a sequence is crucial for determining if a sequence converges or diverges. When we talk about the "limit of a sequence," we are exploring what value the terms of the sequence approach as the sequence progresses towards infinity. If the terms are getting closer and closer to a specific number, the sequence is said to converge to that number, which is the limit. If there's no such number, the sequence diverges. For example, for the sequence given in the problem, \(a_n = \sqrt[n]{4^n n}\), the step-by-step solution shows how we rewrite this sequence using exponent rules. This involves expressing the sequence in a form that makes it easier to see its behavior as \(n\) increases. The rewritten sequence \(a_n = 4 \cdot n^{1/n^2}\) clearly converges, because \(n^{1/n^2}\) approaches 1 as \(n\) approaches infinity. Thus, the limit is 4.

Exponent Rules

Exponent rules are incredibly helpful tools that simplify calculations with powers and roots. These rules include understanding how exponents and roots interact with each other in expressions. One key rule used in the solution is that a root can be expressed as an exponent: \(\sqrt[n]{x} = x^{1/n}\). In the original sequence, \(a_n = \sqrt[n]{4^n n}\), the nth root is expressed as an exponent. By applying exponent rules, this is rewritten as \(a_n = (4^n n)^{1/n}\). Exponentiation rules further allow this to be simplified as \(4 \cdot (n^{1/n})^{1/n}\). With this insight, analyzing the behavior of the sequence becomes much more straightforward, paving the way to evaluate its limit effectively.

Root of a Number

The concept of the root of a number is central to evaluating sequences like the one in the exercise. A root, like the nth root, finds a value which, when multiplied by itself n times, gets you back to the original number. The exercise involves the nth root of \(4^n n\), an expression that initially looks complex. By representing the root in terms of an exponent, \(a_n = (4^n n)^{1/n}\), we see it's possible to apply additional logic with the properties of exponents. Specifically, each part can be addressed using familiar rules: \(4^n\) under the influence of \(1/n\) becomes \(4\) as it simplifies down, and \(n^{1/n}\) approaches 1 as \(n\) approaches infinity. By mastering the root concept alongside exponent rules, intricate sequences become more approachable, making it easier to determine convergence and limits.