Question

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

Short answer

The series diverges by comparison to the harmonic series \( \sum \frac{1}{n} \).

Step-by-step solution

Identify the Test to Use

We will use the Direct Comparison Test, which allows us to determine the convergence of a series by comparing it to a simpler series whose behavior (in terms of convergence or divergence) we already know.

Find a Comparable Series

Observe the terms of the original series: \( \frac{1}{\sqrt{n^2 - 1}} \). For large \( n \), \( n^2 - 1 \approx n^2 \), so \( \sqrt{n^2 - 1} \approx n \). Thus, a comparable series is \( b_n = \frac{1}{n} \).

Analyze the Comparable Series

The series \( \sum_{n=2}^{\infty} \frac{1}{n} \) is a harmonic series, which is known to diverge. To apply the Direct Comparison Test effectively, we need to ensure that \( a_n \geq b_n \) for all sufficiently large \( n \).

Compare the Terms of the Series

For \( n \geq 2 \), consider the inequality \( \sqrt{n^2 - 1} \leq n \), which implies \( \frac{1}{\sqrt{n^2 - 1}} \geq \frac{1}{n} \). Hence, \( a_n = \frac{1}{\sqrt{n^2 - 1}} \geq b_n = \frac{1}{n} \).

Apply the Direct Comparison Test

Since \( \sum_{n=2}^{\infty} \frac{1}{n} \) diverges and \( a_n \geq b_n \), by the Direct Comparison Test, the original series \( \sum_{n=2}^{\infty} \frac{1}{\sqrt{n^2 - 1}} \) also diverges.

Key concepts

Convergence

In mathematics, convergence is an essential concept when dealing with series and sequences. A series converges if the sum of its terms approaches a finite number as the number of terms goes to infinity. When we say a series converges, we are checking if adding infinitely many terms results in a stable number.
In contrast, if the sum of the terms continues to grow without bound, the series is considered divergent. Convergence is crucial because it allows us to understand the behavior of an infinite series and whether it represents a meaningful value.

Harmonic Series

The harmonic series is one of the simplest and most famous divergent series. It is expressed as \[ \sum_{n=1}^{\infty} \frac{1}{n} \] which means you add the reciprocals of all positive integers. Despite its simple form, the harmonic series does not converge.
As more terms are added, the sum grows infinitely large.
This property makes the harmonic series useful in the Direct Comparison Test, as it provides a clear benchmark for divergence. While it may seem counterintuitive, the harmonic series proves that even small terms can eventually add up to an infinite sum.

Series Comparison

Series comparison is a technique used to determine the behavior of one series by comparing it to another series whose convergence or divergence is known. The Direct Comparison Test is one method to do this effectively.
When applying this test, the given series is compared term-by-term with another series. The goal is to establish an inequality between the terms of the two series, such as:

• If a series is smaller than a convergent series, it converges.
• If a series is larger than a divergent series, it diverges.

The choice of comparison series is critical: it must be simpler and its behavior well-known. This helps to make definitive conclusions about the convergence or divergence of the original series.

Divergence

Divergence is the opposite of convergence. A divergent series does not approach a finite limit; instead, its sum grows indefinitely or oscillates without settling on a value. Understanding divergence is fundamental when analyzing infinite series.
One way to demonstrate divergence is through comparison to a known divergent series, such as the harmonic series. In the given exercise, we compared the original series \( \sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2} - 1}} \) to the harmonic series, revealing that it too diverges.
Recognizing divergence is important because it signals that a series does not converge to a meaningful result, which has implications in mathematical models and computations.