Question

Determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{1}{3 n^{3}+7}$$

Short answer

The series converges by the Comparison Test.

Step-by-step solution

Identify the General Term

The general term of the series is given by \(a_n = \frac{1}{3n^3 + 7}\). This is the term we will use to apply the Ratio Test.

Calculate \( \frac{a_{n+1}}{a_n} \)

According to the Ratio Test, we need \( \frac{a_{n+1}}{a_n} \). Compute the expression:\[ \frac{a_{n+1}}{a_n} = \frac{\frac{1}{3(n+1)^3 + 7}}{\frac{1}{3n^3 + 7}} = \frac{3n^3 + 7}{3(n+1)^3 + 7}\]

Simplify and Analyze the Limit

Simplify the expression from the previous step:\[\frac{3n^3 + 7}{3(n+1)^3 + 7} \approx \frac{3n^3 + 7}{3n^3 + 9n^2 + 9n + 10}\]Now, find the limit as \(n \to \infty\):\[\lim_{n \to \infty} \frac{3n^3 + 7}{3n^3 + 9n^2 + 9n + 10} = \lim_{n \to \infty} \frac{3 + \frac{7}{n^3}}{3 + \frac{9}{n} + \frac{9}{n^2} + \frac{10}{n^3}} = 1\]

Conclusion of Ratio Test

The limit of \( \frac{a_{n+1}}{a_n} \) as \(n \to \infty\) is 1. Since the Ratio Test gives a result of 1, it is inconclusive.

Use the Comparison Test for Convergence

Notice that \( \frac{1}{3n^3 + 7} \leq \frac{1}{3n^3} = \frac{1}{3} \cdot \frac{1}{n^3} \). Since \( \sum \frac{1}{n^3} \) is a convergent p-series with \(p = 3 > 1\), by the Comparison Test, the series \( \sum \frac{1}{3n^3 + 7} \) also converges.

Key concepts

Comparison Test

The Comparison Test is a powerful tool to determine the convergence or divergence of series. It works by comparing the series in question to another series, whose convergence properties are well known. To use this test, you must find a series with terms that are always greater than or equal to or always less than or equal to the terms of the series you are investigating. If the series you are comparing to is convergent and your series' terms are smaller, then your series also converges. Conversely, if your series' terms are larger and the comparison series diverges, then your series diverges as well.

In the given exercise, we use the Comparison Test after finding the Ratio Test to be inconclusive. Here, we notice that each term of the series \( \frac{1}{3n^3 + 7} \leq \frac{1}{3n^3} \). The series \( \sum \frac{1}{3n^3} \) is a simpler, similar series that we can handle more easily. Since \( \sum \frac{1}{n^3} \) is known to converge (as a p-series with \( p = 3 \)), the Comparison Test tells us that our original series also converges.

p-series

A p-series is a particular kind of infinite series that takes the form \( \sum \frac{1}{n^p} \), where \( p \) is a positive real number. The convergence of a p-series solely depends on the value of \( p \).

• If \( p > 1 \), the p-series converges.
• If \( p \leq 1 \), the p-series diverges.

For instance, the series \( \sum \frac{1}{n^3} \) has \( p = 3 \), which is greater than 1, hence it converges. In the current exercise, recognizing that the series \( \sum \frac{1}{3n^3} \) is related to \( \sum \frac{1}{n^3} \) helps us determine the convergence of our more complex series by using the Comparison Test. Because \( \sum \frac{1}{n^3} \) converges due to its p-series nature, and the terms of our series are smaller, we conclude that our series converges as well.

Convergence

Convergence is a fundamental concept in the study of series. A series converges when the sum of its infinite terms approaches a finite limit. In other words, as you add more terms, the total sum gets closer and closer to a certain number. Various tests exist for checking convergence, each suitable for different types of series.

The Ratio Test is often used to check convergence, but it can sometimes be inconclusive, as was the case in the provided problem. When such tests fail to clearly determine convergence, other methods such as the Comparison Test are employed. Understanding convergence also helps in assessing whether a series will serve as a valid solution to mathematical problems involving summation of terms.

Series

An infinite series is a sum of terms that continues indefinitely. Mathematically, a series is represented in the form \( \sum_{n=1}^{ ext{∞}} a_n \), where \( a_n \) represents the general term. Calculating the sum of series is a significant part of calculus and analysis.

Some series converge, reaching a finite sum, while others diverge, without approaching any limit. There are various types of series, such as arithmetic series, geometric series, and p-series, each with distinct properties.

To determine whether a series converges or diverges, mathematicians use a variety of test methods, including the Ratio Test and the Comparison Test. Each test has specific applications and is used depending on the structure of the series in question. By employing these methods, we can gain insights into the behavior of series and their sums.