Problem 93 - Algebra and Trigonometry

Question

Limiting Behavior of Powers Complete the following tables. What happens to the \(n\) th root of 2 as \(n\) gets large? What about the \(n\) th root of \(\frac{1}{2} ?\) $$ \begin{array}{|c|c|}\hline n & {2^{1 / n}} \\ \hline 1 & {} \\ {2} & {2} \\\ {5} \\ {10} \\ {100} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|}\hline n & {\left(\frac{1}{2}\right)^{1 / n}} \\ \hline 1 & {} \\ {} & {2} & {} \\ {5} & {5} \\ {10} \\ {100} & {} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

Both \(n\)th roots approach 1 as \(n\) becomes large.

Step 1: Understand the Problem

We are tasked with exploring what happens to \(2^{1/n}\) and \(\left(\frac{1}{2}\right)^{1/n}\) as \(n\) becomes very large. These expressions represent the \(n\)th root of 2 and the \(n\)th root of \(\frac{1}{2}\), respectively.

Step 2: Calculate Values for 2^(1/n)

We'll start filling out the table for \(2^{1/n}\) using different values of \(n\): - For \(n = 1\), \(2^{1/1} = 2\).- For \(n = 2\), \(2^{1/2} = \sqrt{2} \approx 1.414\).- For \(n = 5\), \(2^{1/5} \approx 1.1487\).- For \(n = 10\), \(2^{1/10} \approx 1.0718\).- For \(n = 100\), \(2^{1/100} \approx 1.0069\).

Step 3: Calculate Values for (1/2)^(1/n)

Next, fill out the table for \(\left(\frac{1}{2}\right)^{1/n}\):- For \(n = 1\), \(\left(\frac{1}{2}\right)^{1/1} = \frac{1}{2} = 0.5\).- For \(n = 2\), \(\left(\frac{1}{2}\right)^{1/2} = \frac{1}{\sqrt{2}} \approx 0.707\).- For \(n = 5\), \(\left(\frac{1}{2}\right)^{1/5} \approx 0.8706\).- For \(n = 10\), \(\left(\frac{1}{2}\right)^{1/10} \approx 0.9330\).- For \(n = 100\), \(\left(\frac{1}{2}\right)^{1/100} \approx 0.9931\).

Step 4: Analyze the Limiting Behavior

Observe how both \(2^{1/n}\) and \(\left(\frac{1}{2}\right)^{1/n}\) approach 1 as \(n\) becomes very large. This is because any number raised to a power that approaches zero results in the number approaching 1.

Key Concepts

nth root

When we talk about the nth root of a number, we refer to a root that is derived by dividing the number into "n" equal parts—it's like asking how many times that root needs to be multiplied by itself to get the original number back. For instance, the nth root of 2, written as \(2^{1/n}\), examines how 2 behaves when you take its root a significant number of times. Similarly, \(\left(\frac{1}{2}\right)^{1/n}\) explores a similar scenario but with the fraction \(\frac{1}{2}\).

As we fill out our table with different values of \(n\), such as 1, 2, 5, 10, and 100, we notice a key pattern: the larger the \(n\) value, the closer the nth root of 2 and \(\frac{1}{2}\) get to 1. This behavior is anchored in the essence of root operations themselves—taking larger roots (through increased \(n\)) increasingly balances out the base number towards 1.

limiting behavior

Limiting behavior helps us understand what a function or expression tends toward as its input approaches some value, frequently infinity. In our exercise, we look at powers' limiting behavior as \(n\) grows large. Such exploration offers insight into how numbers behave under extreme conditions, crucial for higher mathematics.

When examining \(2^{1/n}\) and \(\left(\frac{1}{2}\right)^{1/n}\), as \(n\) increases, both start moving towards a value of 1. This is due to the mathematical principle that a number's power heading towards zero naturally adjusts the output to hover closer to 1. In simpler terms, regardless of how large or small a number initially is, its influence weakens as \(n\) increases, drawing it towards an eventual result of 1.

powers of numbers

Powers of numbers, or exponents, are a core part of mathematical operations, representing how many times a number is multiplied by itself. In our task, we explore the nth roots' limiting behavior through powers. The properties of exponents dictate that any real number raised to the power zero results in 1.

Here in our example, we explore what happens as \(n\) increases for both \(2^{1/n}\) and \(\left(\frac{1}{2}\right)^{1/n}\). While initially the numbers—2 and \(\frac{1}{2}\)—present distinct values under smaller n,with increasing \(n\), they yield outcomes approaching 1. This highlights a fascinating aspect of powers: the greater the power (in this case, fractions derived from large \(n\)), the more subdued its impact, revealing a predictable pattern of convergence in terms of limiting behavior. This demonstrates the power of understanding exponents and roots in grasping infinite treatment of sequences.