Question

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{e^{n / 10}}{2^{n}}\right\\}$$

Short answer

Answer: The limit of the sequence \(\left\\{\frac{e^{n / 10}}{2^{n}}\right\\}\) is 0.

Step-by-step solution

Write down the sequence and its ratio

Given the sequence: $$a_n = \frac{e^{n / 10}}{2^{n}}$$ We first find the ratio of consecutive terms: $$\frac{a_{n+1}}{a_{n}} = \frac{\frac{e^{(n + 1) / 10}}{2^{n+1}}}{\frac{e^{n / 10}}{2^{n}}}$$

Simplify the ratio of consecutive terms

We can rewrite and simplify the ratio as: $$\frac{a_{n+1}}{a_{n}} = \frac{e^{(n + 1) / 10}}{2^{n+1}} \cdot \frac{2^n}{e^{n / 10}} = \frac{e^{(n+1)/10 - n/10}}{2}$$ Now, the expression inside the fraction becomes: $$\frac{a_{n+1}}{a_{n}} = \frac{e^{1/10}}{2}$$

Apply Theorem 8.6

Since we have found the ratio of consecutive terms to be constant and equal to \(\frac{e^{1/10}}{2}\), we can now apply Theorem 8.6. By Theorem 8.6, if \(a_n\) converges, then: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{e^{n / 10}}{2^{n}}$$ $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{e^{1/10}}{2}$$ Since the ratio \(\frac{a_{n+1}}{a_n} = \frac{e^{1/10}}{2}\) is constant, the limit of the sequence is determined by the ratio of the sequence. If the ratio is less than 1, the sequence converges, and if the ratio is greater than 1, the sequence diverges. Since \(0<\frac{e^{1/10}}{2}<1\), the sequence \({a_n}\) converges.

Compute the limit

Now that we know the sequence converges, we can find the limit L. Since the limit of the ratio is less than 1, the sequence converges to 0: $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{e^{n / 10}}{2^{n}} = 0$$ So the limit of the given sequence is 0.

Key concepts

Limit of a Sequence

Understanding the limit of a sequence is essential in calculus as it provides a way to describe the behavior of sequences as their terms grow larger. A sequence is simply a list of numbers in a specific order, and the limit is what the sequence's terms get closer to as the sequence progresses.

For example, let's consider a sequence defined by its general term, such as the one given:
\[a_n = \frac{e^{n / 10}}{2^{n}}\]
As n increases, the terms of the sequence can approach a specific value or grow without bounds. If there's a certain number that the terms of the sequence get closer and closer to, we say that the sequence converges to that number, which is called the limit. If the terms do not approach any number but instead continue to grow, we say the sequence diverges. In this case, the sequence converges to 0, largely because the exponential growth in the denominator (powered by 2) eventually outpaces the growth in the numerator.

Ratio Test

The ratio test is a potent tool for determining the convergence or divergence of sequences, particularly those that are formed by the terms of infinite series. It involves taking the limit of the ratio of consecutive terms in the sequence as n approaches infinity. If this ratio is less than 1, the sequence converges; if greater than 1, it diverges; and if it equals 1, the test is inconclusive.

In practice, if you have a sequence a_n, you would look at the limit of a_{n+1} / a_n. For the given sequence a_n = \frac{e^{n / 10}}{2^{n}}, after simplification, this ratio is constant:
\[\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \frac{e^{1/10}}{2}\]
And since e (the base of natural logarithms) is approximately 2.71828, the ratio is decisively less than 1, which is a clear indication of convergence.

Exponential Functions

An exponential function features a constant base raised to a variable exponent. In calculus and beyond, these functions are critical because they model a wide range of real-world phenomena, including population growth and radioactive decay.

The sequence provided in the exercise contains an exponential function in the numerator e^{n / 10}. The number e is an irrational and transcendental number approximated as 2.71828, and it plays a central role in mathematics, particularly in calculus. Its properties make it a natural choice for describing continuous growth or decay processes.

Although exponential functions grow very rapidly, the sequence's denominator is also an exponential function with a faster growth rate due to its base of 2, which causes the overall sequence a_n to converge to zero.

Theorem 8.6

Theorem 8.6 likely refers to a specific mathematical theorem regarding the convergence of sequences or series within the textbook used. In the context of this specific problem, it serves as the foundation for applying the ratio test. Without knowing the exact statement of Theorem 8.6, we can infer that it relates to the behavior and properties of limits in sequences.

For a sequence like a_n where the ratio of consecutive terms a_{n+1}/a_n is constant, Theorem 8.6 can be used to determine convergence. Since our ratio is less than 1, Theorem 8.6 aligns with our findings that the sequence converges to 0.

This theorem emphasizes understanding the conditions under which a sequence will converge to a limit and is a valuable tool for students tackling homework problems involving infinite sequences or series.