Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{2}+4 n+1}}$$

Short answer

The series diverges by the Divergence Test.

Step-by-step solution

Simplify the Series Expression

First, let's analyze the term \(a_n = \frac{n}{\sqrt{n^2 + 4n + 1}}\). Notice that the expression under the square root, \(n^2 + 4n + 1\), resembles a quadratic whose leading term is \(n^2\). This suggests dominance by \(n^2\) in the asymptotic behavior.

Divide by Dominant Term

Divide the numerator and the denominator by \(n\), the largest power of \(n\) in the expression. This results in \(a_n = \frac{1}{\sqrt{1 + \frac{4}{n} + \frac{1}{n^2}}}\). As \(n\) goes to infinity, the terms \(\frac{4}{n}\) and \(\frac{1}{n^2}\) approach zero.

Find the Asymptotic Behavior

As \(n\to\infty\), the expression \(a_n \approx \frac{1}{\sqrt{1 + 0 + 0}} = 1\). Therefore, the series looks similar to the series \(\sum_{n=1}^{\infty} 1\).

Apply the Divergence Test

Using the divergence test, we check if \(\lim_{n\to\infty} a_n = 1 eq 0\). If the limit of a sequence's terms is not zero, the series \(\sum_{n=1}^{\infty} a_n\) must diverge.

Key concepts

Asymptotic Behavior

Asymptotic behavior in mathematics is an essential concept that involves understanding how a function behaves as it approaches a particular limit, often infinity. In the context of series convergence, it helps us analyze the behavior of the terms of the series as their indices become very large.

The initial step in analyzing the series \( \sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2 + 4n + 1}} \) involves simplifying the term \( a_n = \frac{n}{\sqrt{n^2 + 4n + 1}} \). To comprehend its asymptotic behavior, we focus on the dominant terms as \( n \) approaches infinity. The leading term of the denominator, \( n^2 \), indicates that this term's influence becomes more significant with increasing \( n \).

By simplifying \( a_n \) to \( \frac{1}{\sqrt{1 + \frac{4}{n} + \frac{1}{n^2}}} \), it's clear that as \( n \to \infty \), the additional terms \( \frac{4}{n} \) and \( \frac{1}{n^2} \) diminish close to zero, leaving us with an approximate value of 1 for each term. This simplification aligns the series with \( \sum_{n=1}^{\infty} 1 \), an asymptotically equivalent series.

Divergence Test

The Divergence Test, also known as the nth-term test for divergence, is a straightforward method employed in determining whether an infinite series diverges. It is one of the simplest tests because it focuses on the limit of the terms in the series as \( n \to \infty \).

• If \( \lim_{n\to\infty} a_n eq 0 \), the series \( \sum_{n=1}^{\infty} a_n \) automatically diverges.
• If \( \lim_{n\to\infty} a_n = 0 \), the test is inconclusive, and other methods should be considered to determine convergence or divergence.

In our exercise, for the series \( \sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2 + 4n + 1}} \), the asymptotic simplification revealed that \( a_n \approx 1 \). Applying the Divergence Test means evaluating \( \lim_{n\to\infty} a_n = 1 \). Since this limit is not zero, the series indeed diverges.

This decision is critical because determining the non-zero limit of the sequence's terms offers an efficient conclusion that the infinite series does not converge.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Understanding whether an infinite series converges or diverges is central to many areas of mathematics, physics, and engineering.

The concept of convergence refers to the idea that as we add more and more terms of the series, the sum approaches a specific finite value. Conversely, if the series diverges, the sum does not settle toward any particular value as more terms are included.

• Converging series reach a finite number.
• Diverging series grow without bound or oscillate indefinitely.

The given exercise focuses on an infinite series with terms \( \frac{n}{\sqrt{n^2 + 4n + 1}} \). In this context, we applied the Divergence Test and found that the series does not converge because the terms do not tend towards zero. As a result, the series diverges, indicating that its sum will not approach a finite limit. This approach underscores how important asymptotic behavior and limits are in the study of infinite series.