Chapter 1: Problem 3

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=1}^{\infty} \frac{1}{n^{\ln 3}}$$

Short Answer

Expert verified

The series converges by the p-series test because $\ln 3 > 1$.

Step by step solution

Understand the Series

The given series is $\sum_{n=1}^{\text{∞}} \frac{1}{n^{\ln 3}}$. We need to determine whether this series converges or diverges.

Preliminary Test

The preliminary test involves checking whether the terms of the series approach zero as $n \to \text{∞}$. Evaluate $\lim_{{n \to \text{∞}}} \frac{1}{n^{\ln 3}}$. If this limit is not zero, the series diverges immediately.

Evaluate the Limit

Calculate the limit: \ \ $\lim_{{n \to \text{∞}}} \frac{1}{n^{\ln 3}} = \lim_{{n \to \text{∞}}} n^{-\ln 3}$.\ Since $\ln 3$ is a positive constant, $n^{-\ln 3}$ will approach zero as $n \to \text{∞}$. Thus, $ \lim_{{n \to \text{∞}}} \frac{1}{n^{\ln 3}} = 0$.

Select the Appropriate Test

Recognize that the series has the form $\sum_{n=1}^{\text{∞}} \frac{1}{n^p}$. This suggests using the p-series test.

Apply the p-series Test

In a p-series $\sum_{n=1}^{\text{∞}} \frac{1}{n^p}$, the series converges if $p > 1 $ and diverges if $p \leq 1$. Here, $p = \ln 3$. Since $\ln 3 \approx 1.0986$, which is greater than 1, the series converges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

series convergence tests

When working with series, it's essential to determine whether the series converges or diverges. This analysis helps us understand if the sum of the infinite series reaches a finite value or keeps growing indefinitely. Several tests can be applied to decide this.

The preliminary test is usually the first step. It checks whether the limit of the terms of the series approaches zero as the number of terms increases infinitely. If the terms do not approach zero, the series diverges automatically.

Beyond the preliminary test, there are several more specific tests. These include the p-series test, the ratio test, the root test, and the comparison test. Each has its criteria and can be useful depending on the series in question. Let's dive into the p-series test and the preliminary test in more detail.

p-series test

The p-series test is a straightforward and widely used test for determining the convergence or divergence of series in the form $ \sum_{{n=1}}^{{\text{∞}}} \frac{{1}}{{n^p}} $.

According to the p-series test, the series converges if the exponent $ p $ is greater than 1 and diverges if $ p \leq 1 $. For instance:

If $ p = 2 $, the series $ \sum_{{n=1}}^{{\text{∞}}} \frac{{1}}{{n^2}} $ converges.
If $ p = 0.5 $, the series $ \sum_{{n=1}}^{{\text{∞}}} \frac{{1}}{{n^{0.5}}} $ diverges.

In the given series example, we have $ \sum_{{n=1}}^{{\text{∞}}} \frac{{1}}{{n^{\text{{ln}} 3}}} $. Here, $ p = \text{{ln}} 3 $, which is approximately 1.0986. Since this value is greater than 1, the series converges by the p-series test. Understanding and applying the p-series test can make analyzing series simpler and more efficient.

preliminary test

The preliminary test is often the first method to apply when determining the convergence of a series. This test, also known as the nth-term test for divergence, involves analyzing the limit of the terms of the series as $ n $ tends to infinity.

The preliminary test states that if the limit of the terms $ a_n $ is not zero, then the series $ \sum a_n $ diverges. Mathematically, this is expressed as: \[ \text{{If }} \lim_{{n \to \text{∞}}} a_n \eq 0, \text{{ then }} \sum a_n \text{{ is divergent}}. \]

In the given problem, the terms of the series are $ \frac{{1}}{{n^{{\text{ln}} 3}}} $. We calculate the limit as $ n $ approaches infinity. Since $ \lim_{{n \to \text{∞}}} n^{-{\text{ln}} 3} = 0 $, we cannot conclude divergence from the preliminary test alone. This means we need additional tests, like the p-series test, to properly conclude the behavior of the series. The preliminary test, though simple, is a powerful starting point in the series analysis process.