Question

Use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}+3 n-5}$$

Short answer

The series converges by comparison with the convergent \( p \)-series \( \sum \frac{1}{n^2} \).

Step-by-step solution

Identify the General Term of the Series

The series is given as \( \sum_{n=1}^{\infty} \frac{1}{n^{2}+3n-5} \). The general term of this series is \( a_n = \frac{1}{n^{2}+3n-5} \).

Compare with a Known Series

To use the Direct Comparison Test, we need to compare our series with a series that we already know converges or diverges. A useful series for comparison is the \( p \)-series given by \( \sum \frac{1}{n^p} \), which converges if \( p > 1 \). We will compare \( a_n \) with \( b_n = \frac{1}{n^2} \) since for large \( n \), \( n^2 + 3n - 5 \approx n^2 \).

Set Up the Inequality for Comparison

We need to demonstrate that \( a_n \leq b_n \) for all \( n \) sufficiently large. Note that \( a_n = \frac{1}{n^{2}+3n-5} \leq \frac{1}{n^2} = b_n \) because \( n^2 \leq n^{2}+3n-5 \) for all \( n \) large enough (specifically \( n > 5 \)).

Use the Direct Comparison Test

Since \( a_n \leq b_n \) for all \( n \) sufficiently large, and we know that \( \sum \frac{1}{n^2} \) converges (it's a \( p \)-series with \( p=2 \)), by the Direct Comparison Test, the series \( \sum \frac{1}{n^{2}+3n-5} \) also converges.

Key concepts

Convergence of Series

Understanding the convergence of series is crucial in mathematics, especially when dealing with infinite processes. A series is said to converge if the sum of its infinite terms approaches a specific value, a finite number, as more terms are added. This property is essential because it allows us to handle infinite series in practical applications and mathematical analyses.

Several tests exist to determine whether a series converges. One such method is the Direct Comparison Test, which involves comparing the series in question to another series, the convergence properties of which are already known. This comparison provides insight into the behavior of the original series, helping to establish whether it converges under certain conditions.

P-Series

The p-series is a specific type of infinite series used frequently in convergence tests. It is expressed in the form \( \sum \frac{1}{n^p} \), where \( p \) is a positive real number, and \( n \) represents the natural numbers starting from 1. The behavior of a p-series is determined by the value of \( p \).

• When \( p > 1 \), the p-series converges. This means that the sum of its terms approaches a finite value as more terms are added.
• If \( p \leq 1 \), the p-series diverges, which means the sum grows without bound with the addition of more terms.

In the context of the Direct Comparison Test, a p-series with known behavior, such as \( \sum \frac{1}{n^2} \) where \( p=2 \), often serves as a benchmark for comparing and determining the convergence of other related series.

Inequalities

Inequalities play a pivotal role in convergence tests, including the Direct Comparison Test. When determining whether a series converges, it’s important to show that the terms of the series in question are smaller than or equal to the terms of a reference series that is known to converge.

In the exercise given, we demonstrated the inequality \( \frac{1}{n^2 + 3n - 5} \leq \frac{1}{n^2} \) for sufficiently large values of \( n \). This inequality stems from the relationship between the denominators \( n^2 \) and \( n^2 + 3n - 5 \). For large values of \( n \), \( n^2 + 3n - 5 \) approaches \( n^2 \), allowing us to establish that

• \( n^2 \leq n^2 + 3n - 5 \)
• Consequently, \( \frac{1}{n^2 + 3n - 5} \leq \frac{1}{n^2} \)

Using this inequality, we can conclude that the given series converges by the Direct Comparison Test.

Infinite Series

Infinite series are sums of an infinite sequence of terms. These series can exhibit convergence or divergence, and they are common in various fields in mathematics and applied sciences. Understanding their behavior is key to analyzing complex mathematical and applied problems.

The Direct Comparison Test is a handy tool when dealing with infinite series. By comparing a given series to another series with known behavior, we can make conclusions regarding its convergence or divergence. Such comparisons must be made carefully, using inequalities to ensure that all necessary conditions for convergence are met.

Infinite series are often used to approximate functions, describe physical phenomena, and solve equations in which there are infinitely many interactions or terms involved, making them an indispensable part of mathematical analysis.