Question

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $$ a_{n}=(2 n)^{1 / n}-n^{1 / n} $$

Short answer

The sequence converges to 0.

Step-by-step solution

Simplifying the Sequence

Given the sequence \( a_n = (2n)^{1/n} - n^{1/n} \), we start by simplifying it. Notice that \( (2n)^{1/n} = 2^{1/n} n^{1/n} \). Therefore, \( a_n = 2^{1/n} n^{1/n} - n^{1/n} = n^{1/n} (2^{1/n} - 1) \).

Analyzing \( n^{1/n} \) as \( n \to \infty \)

As \( n \to \infty \), \( n^{1/n} \) approaches 1. This is because the \( n \)-th root of \( n \) gets closer to 1 as \( n \) becomes larger.

Analyzing \( 2^{1/n} \) as \( n \to \infty \)

Similarly, as \( n \to \infty \), \( 2^{1/n} \) also approaches 1. The \( n \)-th root of any constant number approaches 1 as \( n \) increases. Thus, \( 2^{1/n} - 1 \) approaches 0.

Finding the Limit of \( a_n \)

Both \( n^{1/n} \) and \( 2^{1/n} - 1 \) approach 1 and 0, respectively, as \( n \to \infty \). Therefore, \( a_n = n^{1/n} (2^{1/n} - 1) \to 1 \cdot 0 = 0 \). Thus, the sequence converges to 0.

Key concepts

Limit of a sequence

Understanding the limit of a sequence is essential in determining whether a sequence converges or diverges. A limit is essentially the value that the terms of a sequence approach as the sequence progresses towards infinity.

For the sequence in our exercise, given by \( a_n = (2n)^{1/n} - n^{1/n} \), we are interested in what happens as \( n \) becomes very large. To understand this, we break down the expression:

• We first notice that \( (2n)^{1/n} = 2^{1/n} n^{1/n} \), which we reformulate as \( n^{1/n} (2^{1/n} - 1) \).
• As \( n \to \infty \), it's useful to identify that both \( n^{1/n} \) and \( 2^{1/n} \) approach 1.

Thus, the sequence \( a_n \) can be expected to approach 0, showing its convergence.

Convergence

Convergence is when the terms of a sequence become arbitrarily close to a specific value, known as the limit, as the sequence progresses.

For convergence, it’s essential to demonstrate that the difference between the terms of the sequence and the limit can be made as small as desired by choosing a sufficiently large index.

In our exercise, we find that both components \( n^{1/n} \) and \( 2^{1/n} - 1 \) are crucial:

• As \( n \to \infty \), \( n^{1/n} \to 1 \), indicating each term gets closer to 1.
• And \( 2^{1/n} - 1 \to 0 \), further reducing the terms to zero eventually.

Since these terms multiply to form \( a_n \), and one approaches 0, the entire sequence thus converges to 0. This steady approach to zero confirms the sequence’s convergence.

Mathematical analysis

Mathematical analysis provides the foundation to rigorously study limits and convergence of sequences. It involves breaking down and simplifying complex mathematical expressions to understand their behavior under various conditions.

In the sequence \( a_n = (2n)^{1/n} - n^{1/n} \), analysis involves:

• Identifying the dominant behavior as \( n \to \infty \). This often involves recognizing patterns and characteristics that dictate the sequence's long-term behavior.
• Using concepts like \( n^{1/n} \) and \( 2^{1/n} \) and understanding their individual limits aid in predicting the behavior of the entire sequence.

Through mathematical analysis, we conclude that \( a_n \to 0 \) as \( n \to \infty \). This ability to decompose and analyze components is what makes mathematical analysis such a powerful tool in understanding sequences and series.

With these fundamentals, students can apply similar reasoning to understand the behavior of other sequences.