Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+\cdots\)

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Identify the series

The given series is \( \sum_{n=1}^{\infty} \frac{n}{3^n} \). It is an infinite series where each term is in the form of \( \frac{n}{3^n} \).

Step 2: Consider applicable tests

For this series, we can consider using the Ratio Test to determine convergence or divergence. The Ratio Test is particularly useful for series with factorials, exponentials, or combinations.

Step 3: Apply the Ratio Test

The Ratio Test involves taking the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{n}{3^n} \).Find \( a_{n+1} = \frac{n+1}{3^{n+1}} \) and compute:\[\frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{3^{n+1}}}{\frac{n}{3^n}} = \frac{n+1}{3n}.\]

Step 4: Take the limit of the ratio

Calculate the limit as \( n \to \infty \):\[L = \lim_{n \to \infty} \frac{n+1}{3n} = \lim_{n \to \infty} \frac{n}{3n} + \frac{1}{3n} = \frac{1}{3} + 0 = \frac{1}{3}.\]

Step 5: Examine the result of the Ratio Test

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

Key Concepts

Ratio Test

In mathematics, to determine whether an infinite series converges or diverges, we often use the Ratio Test. It's particularly helpful for series with terms that involve factorials or exponentials. Let's break it down:

• First, you identify the general term of the series, noted as \( a_n \).
• Find the next term, \( a_{n+1} \).
• The core of the Ratio Test is computing the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

When you calculate \( L \):

• If \( L < 1 \), the series converges.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the test is inconclusive; try other tests.

This test simplifies the study of series, especially when terms become complex with increased \( n \). For the example series, the Limit \( L = \frac{1}{3} \) indicated convergence since it was less than 1.

Infinite Series

An infinite series is simply the sum of infinitely many terms. It is typically written as \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) represents a term in the series.

Understanding infinite series is crucial, as they appear frequently in mathematical applications and real-world problems. Here's what to keep in mind:

• The series may either converge (approach a specific value) or diverge (grow without bounds or oscillate).
• Not all infinite series have an easy answer. We use various convergence tests to check their behavior.
• An example of a convergent series is the geometric series, where each term is a constant times the previous term.

Studying specific examples and utilizing different tests help in getting a good grasp of infinite series behavior. It can be a rewarding yet complex topic for learners to master.

Convergence Tests

Convergence tests are techniques that help us determine whether an infinite series converges or diverges. Each test has conditions under which it works best, giving students a toolkit for analysis.

Here's a look at some popular convergence tests besides the Ratio Test:

• Divergence Test: If \( \lim_{n \to \infty} a_n eq 0 \) or doesn't exist, the series diverges.
• Integral Test: Relates the series to an integral to decide convergence.
• Comparison Test: Compares two series to decide the behavior of one based on the other.
• Alternating Series Test: Used for series whose terms alternate in sign.
• Root Test: Similar to the Ratio Test but uses roots instead of ratios.

Each test has its advantages and is chosen based on the specific form and properties of the series in question. Understanding when and how to apply these tests is key to mastering series convergence analysis.