Problem 13 - Thomas Calculus

Question

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}}\end{equation}

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}} \) diverges.

Step 1: Identify the Series

Consider the series \( \sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}} \). We need to find the behavior of this series using the Limit Comparison Test.

Step 2: Simplify the Series

Rewrite the series' term \( a_n \) as follows: \( a_n = \frac{5^{n}}{\sqrt{n} 4^{n}} = \left( \frac{5}{4} \right)^n \cdot \frac{1}{\sqrt{n}} \).

Step 3: Choose a Comparison Series

Compare \( a_n = \left( \frac{5}{4} \right)^n \cdot \frac{1}{\sqrt{n}} \) with \( b_n = \left( \frac{5}{4} \right)^n \). The sequence \( b_n \) is a geometric series with ratio greater than 1, which diverges.

Step 4: Apply the Limit Comparison Test

Calculate \( \lim_{n \to \infty} \frac{a_n}{b_n} \):\[\lim_{n \to \infty} \frac{\left( \frac{5}{4} \right)^n \cdot \frac{1}{\sqrt{n}}}{\left( \frac{5}{4} \right)^n} = \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0.\]Since the limit is zero and \( b_n \) diverges, we cannot use the Limit Comparison Test in this form as is.

Step 5: Analyze the Test Result

The Limit Comparison Test is inconclusive when the limit is zero. However, since \( \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0 \), it indicates \( a_n \) is negligible compared to \( b_n \) which diverges.

Step 6: Conclusion on the Series Behavior

Since we didn't get a clear result from the Limit Comparison Test, consider the individual growth rates: \( \frac{5^n}{4^n} = \left( \frac{5}{4} \right)^n \) grows faster than \( \frac{1}{\sqrt{n}} \) approaches zero, reinforcing the divergence of the series.

Step 7: Final Convergence Judgment

Thus, \( \sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}} \) diverges as the terms \( a_n \) grow without bound compared to the rate by which \( \frac{1}{\sqrt{n}} \) approaches zero.

Key Concepts

Series Convergence

Series convergence is an important concept in calculus that helps us understand whether the sum of all terms in an infinite series results in a finite value or not. When we talk about a series, we are essentially summing up an endless list of numbers. To determine if this sum ever settles at a specific value, we use tests like the Limit Comparison Test. Convergent series are those in which the sum of all terms approaches a fixed number as the series progresses. If the terms get smaller quickly enough, the series eventually stops growing and settles down at a limit.

• A series converges if the sum approaches a finite number.
• If the sum continues to grow without settling, it's divergent.
• The convergence test often involves comparing the series to a well-known convergent or divergent series.

Divergent Series

When we talk about divergent series, we're referring to series that do not have a finite limit. These series either grow indefinitely or oscillate without settling at a particular value. In mathematical terms, a divergent series is one where the limits of the sums do not exist or are infinite. This means that as you add more terms, the total keeps changing without ever settling into a steady number.

• A divergent series grows infinitely or oscillates without converging.
• Terms of a divergent series don't decrease quickly enough to reach a finite sum.
• Identifying divergence often involves proving growth patterns that exceed convergence thresholds.

Geometric Series

A geometric series is a specific type of series where each term is a constant multiple of the previous one. This constant is known as the common ratio. Understanding geometric series is crucial because they have a distinct pattern that allows us to easily determine whether they converge or diverge.If the absolute value of the common ratio is less than 1, the series converges. If it's greater than or equal to 1, the series will diverge.

• A geometric series has the form: \( a + ar + ar^2 + ar^3 + \ldots \), where \( r \) is the common ratio.
• Convergent geometric series have \( |r| < 1 \), meaning they reach a finite sum.
• Divergent geometric series have \( |r| \geq 1 \), leading to an infinite or undefined sum.