Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\left(1+\frac{2}{n}\right)^{n / 2}\)

Step-by-Step Solution

Answer

The sequence converges to \(e\).

Step 1: Find the First Term

Substitute \(n = 1\) into the explicit formula. \[ a_1 = \left(1 + \frac{2}{1}\right)^{1/2} = \sqrt{3} \]

Step 2: Find the Second Term

Substitute \(n = 2\) into the explicit formula. \[ a_2 = \left(1 + \frac{2}{2}\right)^{2/2} = \sqrt{2} \]

Step 3: Find the Third Term

Substitute \(n = 3\) into the explicit formula. \[ a_3 = \left(1 + \frac{2}{3}\right)^{3/2} = \left(\frac{5}{3}\right)^{3/2} \]

Step 4: Find the Fourth Term

Substitute \(n = 4\) into the explicit formula. \[ a_4 = \left(1 + \frac{2}{4}\right)^{4/2} = \left(\frac{3}{2}\right)^2 \]

Step 5: Find the Fifth Term

Substitute \(n = 5\) into the explicit formula. \[ a_5 = \left(1 + \frac{2}{5}\right)^{5/2} \]

Step 6: Analyze Sequence Convergence

To determine convergence, consider the limit: \(\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{n/2}\). This resembles the exponential limit form \((1+\frac{x}{n})^n \rightarrow e^x\) as \(n \to \infty\). Therefore, in our case, \[ \lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{n/2} = e^{2/2} = e \]

Step 7: Conclusion

The first five terms of the sequence are \(\sqrt{3}, \sqrt{2}, \left(\frac{5}{3}\right)^{3/2}, \left(\frac{3}{2}\right)^2, \left(1+\frac{2}{5}\right)^{5/2}\). The sequence converges, and the limit is \(e\).

Key Concepts

Sequence Convergence

To understand sequence convergence, imagine tracing the path of a bouncing ball settling down. Just like how the ball eventually comes to rest, a convergent sequence steadily approaches a particular value, called the limit. In mathematical terms, when the terms of a sequence get closer and closer to a single number as the sequence progresses, we say it converges.

• A sequence is considered convergent if it approaches a specific limit as it extends to infinity.
• The opposite of this is a divergent sequence, which doesn't approach any particular value.

In the original exercise, the sequence given is \[ a_n=\left(1+\frac{2}{n}\right)^{n/2} \]. By analyzing the behavior of this sequence as \( n \) grows, we see that the terms approach \( e \). Therefore, the sequence is convergent, meaning it comes closer to the number \( e \) as \( n \) gets very large.

Limit of a Sequence

The limit of a sequence helps identify the ultimate number a sequence approaches. Establishing the limit is like answering a question about the long-term behavior of a sequence. The limit exists if, after a certain point, all sequence terms become indistinguishably close to some specific number.
To find the limit of the sequence \( a_n=\left(1+\frac{2}{n}\right)^{n/2} \), we relate it to the exponential form \((1 + \frac{x}{n})^n\) which converges to \(e^x\) as \(n\) approaches infinity. Using this principle, we rewrite the sequence limit as \(e^{2/2} = e\). Hence, the sequence converges to \(e\).

• Analyzing the limit of a sequence is crucial to determine if a sequence converges or not.
• The process involves examining the behavior of the terms as \( n \) tends to infinity.

Exponential Function

The exponential function, denoted as \( e^x \), is a powerful tool in mathematics. This function is defined as the limit of \((1 + \frac{x}{n})^n\) as \( n \to \infty \). It's crucial in understanding growth processes and compound-interest calculations.
An interesting property of the exponential function is its connection to limits of sequences, much like the one in the problem. The sequence \( \left(1+\frac{2}{n}\right)^{n/2} \) reflects this concept, approaching \( e \) by employing these foundational exponential properties.

• Exponential functions have applications in growth models across different fields.
• The foundational connection between sequences and \( e \) arises from limits related to the exponential function.

The beauty of the exponential function lies in how it connects limits, sequences, and growth.