Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1}\)

Step-by-Step Solution

Answer

The series converges by the Limit Comparison Test.

Step 1: Analyze the Series

We begin by analyzing the given infinite series \( \sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1} \). Our goal is to determine whether this series converges or diverges. The terms of this series are non-negative, which allows us to consider convergence tests applicable to series with non-negative terms.

Step 2: Use the Limit Comparison Test

The Limit Comparison Test is often useful for rational functions and involves comparing the given series to a known series. In this case, we compare it with \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \), a p-series with \( p = \frac{3}{2} \) that is known to converge.

Step 3: Compute the Limit for Comparison

We calculate the limit \( \lim_{n \to \infty} \frac{\frac{\sqrt{n+1}}{n^2+1}}{\frac{1}{n^{3/2}}} = \lim_{n \to \infty} \frac{\sqrt{n+1} \cdot n^{3/2}}{n^2+1} \). As \( n \to \infty \), this simplifies to \( \lim_{n \to \infty} \frac{n^{3/2} \cdot \sqrt{n}}{n^2} = \lim_{n \to \infty} \frac{n^2}{n^2} = 1 \). Thus, the limit is a positive finite number.

Step 4: Conclude the Convergence Using the Limit Comparison Test

Since the limit is a positive finite number and the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) converges, by the Limit Comparison Test, the series \( \sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1} \) also converges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool in determining the convergence of an infinite series. It is particularly helpful when dealing with series where terms have complicated expressions. To apply the Limit Comparison Test, we choose another series whose behavior (in terms of convergence or divergence) is already known. By comparing our series to this benchmark, we can make conclusions about convergence.

Here is how it works:

• Select a known series that closely resembles the one you are analyzing. For rational functions, a p-series is often chosen because they are well understood.
• Form the limit of the ratio of the terms of the given series and the chosen series as \( n \to \infty \).
• If this limit is a finite positive number, then both series either converge or diverge together.

In our exercise, we compared \( \sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1} \) to the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). As their limit was positive and finite, it confirmed that our series converges.

P-Series

A p-series is a series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a positive constant. Understanding the convergence properties of p-series is key, because it often comes up in infinite series analysis.

Key insights of p-series convergence:

• If \( p > 1 \), the series converges, meaning the sum reaches a finite limit.
• If \( p \leq 1 \), the series diverges, which means the sum grows indefinitely.

In our solved step, we employed the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) because \( p=\frac{3}{2} > 1 \), and thus this series converges. Such an understanding helped in applying the Limit Comparison Test effectively.

Rational Functions

Rational functions are expressions that include ratios of polynomials. In infinite series analysis, when terms are given as rational functions, it is commonly beneficial to simplify or approximate them, especially for large \( n \). This helps apply suitable tests for convergence.

Our series, \( \sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{2}+1} \), incorporates a rational function. We simplify the series by focusing on dominant terms when \( n \) is large. The numerator behaves like \( \sqrt{n} \) and the denominator as \( n^2 \), making it closely related to \( \frac{1}{n^{3/2}} \), a familiar form in convergence tests.Simplifying these ratios ensures the correct application of the Limit Comparison Test, leading to accurate conclusions.

Infinite Series Analysis

Infinite series analysis seeks to determine the behavior of series whose terms extend indefinitely. Tools like the Limit Comparison Test, understanding of rational functions, and properties of p-series are essential in this analysis.

A good starting point is identifying the form of the infinite series and notating the essential characteristics, such as non-negative terms. This allows us to choose suitable convergence tests. By analyzing the behavior of a series as \( n \to \infty \), we often approximate terms to more manageable forms.

For instance:

• Simplification via dominant terms allows us to use similar known convergent series to draw conclusions.
• Efficient application of the Limit Comparison Test enables a clear understanding of convergence or divergence.

Such strategies not only help in exercises but also in comprehending more complex series in higher mathematics.