Problem 17 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt {n^2 + 1}} \)

Step-by-Step Solution

Answer

The series diverges by the Limit Comparison Test.

Step 1: Identify the Series Type

The given series is \( \sum_{n = 1}^{\infty} \frac{1}{\sqrt{n^2 + 1}} \). This appears to be a series where each term is of the form \( a_n = \frac{1}{\sqrt{n^2 + 1}} \). This series does not clearly fit a geometric or p-series form, so we will need to use a convergence test.

Step 2: Use the Limit Comparison Test

For the Limit Comparison Test, we compare our series with a simpler series that we know the behavior of. We choose \( b_n = \frac{1}{n} \), because for large \( n \), \( \sqrt{n^2 + 1} \approx n \). Thus, \( a_n \approx \frac{1}{n} \), a harmonic series which diverges.

Step 3: Calculate the Limit for Comparison

We calculate the limit:\[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{\sqrt{n^2 + 1}}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{\sqrt{n^2 + 1}}. \]

Step 4: Evaluate the Limit

Simplify and evaluate the limit: \[ \lim_{n \to \infty} \frac{n}{\sqrt{n^2 + 1}} = \lim_{n \to \infty} \frac{1}{\sqrt{1 + \frac{1}{n^2}}} = \frac{1}{\sqrt{1 + 0}} = 1. \]

Step 5: Conclude with Limit Comparison Test

Since the limit is finite and non-zero, specifically \( 1 \), and \( \sum \frac{1}{n} \) diverges, by the Limit Comparison Test, the series \( \sum \frac{1}{\sqrt{n^2 + 1}} \) must also diverge.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool for determining the convergence or divergence of a series when direct observation is challenging.

When faced with a complex series, sometimes it's beneficial to compare it with a simpler one whose behavior (convergence or divergence) is already known. The essence of the Limit Comparison Test is to determine how the given series behaves relative to a known benchmark series.

• Identify a known benchmark series \( b_n \) to compare with your series \( a_n \).
• Calculate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If the limit is a positive finite number, then both series either converge or diverge together.

In our context, we compared \( a_n = \frac{1}{\sqrt{n^2+1}} \) with \( b_n = \frac{1}{n} \), a harmonic series. By setting the limit of their ratio to 1, we established that they share the same behavior—both diverge.

Harmonic Series

A harmonic series is a classic example of a divergent series. It is defined as\[ \sum_{n=1}^{\infty} \frac{1}{n}, \]where each successive term is the reciprocal of an integer. The harmonic series grows very slowly but ultimately diverges, meaning it doesn't sum to a finite number as n approaches infinity. This is counterintuitive because at face value, each term seems to get smaller, yet they don't "shrink" fast enough to sum up to a finite limit.In practice:

• The divergence of the harmonic series is crucial for testing other complex series.
• It's a standard reference series when employing the Limit Comparison Test.

Understanding the divergence of the harmonic series helps grasp why our original problem \( \sum \frac{1}{\sqrt{n^2 + 1}} \) also diverges, as it closely mimics the behavior of the harmonic series.

Series Convergence Tests

Series convergence tests offer a variety of methods to evaluate whether a series will converge or diverge. These tests apply different criteria depending on the type of series in question. Common types include:

• Geometric Series Test
• \( p \)-Series Test
• Comparison Tests (Direct and Limit)
• Ratio Test
• Root Test

Each method has its own rationality and application scenarios. For example, the \( p \)-Series Test helps when terms are structured like \( \frac{1}{n^p} \), while the Limit Comparison Test is beneficial when a direct form is not clear.In our exercise, the Limit Comparison Test was most suitable because the series did not directly fit the geometric or \( p \)-series forms. Remember, the choice of test is vital to efficiently determining convergence or divergence.