Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Ratio Test to determine convergence or divergence. $$ \sum_{n=1}^{\infty} n\left(\frac{1}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Write the General Term

Identify the general term of the series, which is given by \( a_n = n \left( \frac{1}{3} \right)^n \).

Step 2: Find the Next Term

Write the next term of the series as \( a_{n+1} = (n+1) \left( \frac{1}{3} \right)^{n+1} \).

Step 3: Set Up the Ratio Test Expression

Compute the ratio \( \frac{a_{n+1}}{a_n} \):\[\frac{a_{n+1}}{a_n} = \frac{(n+1) \left( \frac{1}{3} \right)^{n+1}}{n \left( \frac{1}{3} \right)^n} = \frac{n+1}{n} \cdot \frac{1}{3}\].

Step 4: Simplify the Expression

Simplify the expression obtained: \[ \frac{n+1}{n} \cdot \frac{1}{3} = \frac{1}{3} \cdot \left( 1 + \frac{1}{n} \right) \].

Step 5: Take the Limit as n Approaches Infinity

Calculate the limit \( L \) of the simplified expression as \( n \to \infty \):\[ L = \lim_{n \to \infty} \frac{1}{3} \left( 1 + \frac{1}{n} \right) = \frac{1}{3} \times 1 = \frac{1}{3}\].

Step 6: Interpret the Result with the Ratio Test

Since \( L = \frac{1}{3} < 1 \), by the Ratio Test, the series \( \sum_{n=1}^{\infty} n \left( \frac{1}{3} \right)^n \) converges.

Key Concepts

convergence and divergence

The concepts of convergence and divergence are crucial when dealing with infinite series. An infinite series is a sum of an infinite sequence of terms. But we need to know whether this sum results in a finite value (convergence) or not (divergence). In simpler terms:

• A series converges if the sum of its terms approaches a specific number as more terms are added.
• A series diverges if the sum keeps increasing or decreasing without bound.

For instance, think of convergence like filling a glass with water. Once full, no matter how much more water you add, it still can't hold more. It has reached a limit. Conversely, a diverging series is like pouring water into a bottomless pit. No matter how much you pour, it never fills up.

infinite series

An infinite series is a mathematical expression where an infinite number of terms are added together. Let's consider:\[\sum_{n=1}^{\infty} a_n\]Where \(a_n\) represents the general term of the series. This indicates that starting from the first term, you're adding an infinite number of subsequent terms. While it sounds overwhelming, infinite series are critical in mathematics, especially in calculus and analysis.Analyzing infinite series involves determining whether they converge or diverge. This is where tests like the Ratio Test come into play. The Ratio Test is particularly effective for series where each term is a product of a constant and a variable raised to a power, like exponential terms.Working with infinite series requires a firm understanding of limits since convergence is often about approaching a specific limit as more terms are added.

general term

The general term \(a_n\) defines each term in a sequence or series. It's a mathematical expression that helps in identifying all individual terms of the series.For the given series:\[a_n = n \left(\frac{1}{3}\right)^n\]This means that the nth term is calculated by multiplying \(n\) (the position of the term) by \(\left(\frac{1}{3}\right)^n\), the fraction raised to the power of \(n\). It's essential because, without the general term, it would be impossible to analyze the behavior of the entire series.By manipulating the general term, you can derive various aspects such as the arrangement of terms and observe patterns that help in concluding about convergence or divergence. To find the next term or explore specific characteristics of the sequence, the general term is your foundational tool. It's like having the recipe for a dish you're endlessly repeating, but with potentially different outcomes each time based on the ingredients in your kitchen (the position \(n\)).