Question

Use the integral test to determine whether the following sums converge. \(\sum_{n=1}^{\infty} \frac{2 n}{1+n^{4}}\)

Short answer

The series \( \sum_{n=1}^{\infty} \frac{2n}{1+n^{4}} \) converges by the integral test.

Step-by-step solution

Understand the Integral Test

The integral test states that if \( f(x) \) is a continuous, positive, decreasing function for \( x \geq 1 \), and \( a_n = f(n) \), then the series \( \sum_{n=1}^{\infty} a_n \) and the integral \( \int_{1}^{\infty} f(x) \, dx \) either both converge or both diverge.

Define the Function for the Integral Test

In this problem, we consider \( a_n = \frac{2n}{1+n^4} \) and define \( f(x) = \frac{2x}{1+x^4} \). We ensure \( f(x) \) is continuous, positive, and decreasing for \( x \geq 1 \).

Check if the Function Decreases

To check if \( f(x) \) decreases, compute its derivative: \( f'(x) = \frac{d}{dx} \left( \frac{2x}{1+x^4} \right) \). Using the quotient rule, we find \( f'(x) = \frac{2(1+x^4)-8x^4}{(1+x^4)^2} = \frac{2 - 6x^4}{(1+x^4)^2} \). For \( x \geq 1 \), \( f'(x) < 0 \), indicating \( f(x) \) is decreasing.

Set Up the Improper Integral

We need to evaluate the improper integral \( \int_{1}^{\infty} \frac{2x}{1+x^4} \, dx \). If this integral converges, so does the series.

Use Substitution to Evaluate the Integral

Apply the substitution \( u = x^2 \), which gives \( du = 2x \, dx \). The integral becomes \( \int \frac{1}{1+u^2} \, du \). This is the arctangent function, so we have \( \arctan(u) \). Thus, the integral is \( \arctan(x^2) \) evaluated from \( x = 1 \) to \( x = \infty \).

Determine Convergence of the Integral

Evaluate \( \lim_{b \to \infty} (\arctan(b^2) - \arctan(1)) \). Since \( \arctan(x^2) \to \frac{\pi}{2} \) as \( x \to \infty \), the integral \( \arctan(\infty) - \arctan(1) \) converges.

Conclude About the Series Convergence

Since the integral converges, by the integral test, the series \( \sum_{n=1}^{\infty} \frac{2n}{1+n^4} \) also converges.

Key concepts

Series Convergence

In mathematics, determining whether a series converges is a key concept when analyzing infinite sums. A series is an expression of the form \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) represents a term in the sequence.
Convergence signifies that as we add up the terms, the series approaches a finite number, known as the series' limit. Understanding convergence helps us know whether summing infinitely many numbers results in something meaningful.
To decide if a series converges, several tests can be applied, such as the Integral Test, which is particularly useful when dealing with functions related to continuous series terms.
By using tools like the Integral Test, we can determine whether a given series will eventually settle on a specific value or diverge towards infinity, making it critically helpful in more complex analyses.

Improper Integral

Improper integrals help evaluate the behavior of functions over infinite intervals or discontinuous areas. In calculus, these integrals extend beyond the traditional bounds.
An improper integral such as \( \int_{1}^{\infty} f(x) \, dx \) evaluates the area under the curve \( f(x) \) from some point to infinity. The concept hinges on taking limits: as we extend the bounds of the integral towards infinity, we check if the area converges to a finite value.
Solving these integrals involves substitution or transformation techniques, like turning an infinite domain into one we can comprehend. Proper evaluation often involves recognizing known results, such as the limit behavior of arctangent in this problem.
A converging improper integral often represents the success of translating endless, complex phenomena into finite, comprehensible outcomes, mirroring the convergence principles used in series.

Continuous Functions

Continuous functions form the backbone of calculus and analysis, being vital for applying the Integral Test for series convergence. A function is continuous if there are no sudden jumps or breaks – it can be graphed without lifting a pen.
For example, a continuous function \( f(x) \) over an interval ensures no surprises when applying calculus tools such as integration and differentiation.
In analyzing series, continuity is paramount, as it guarantees that the Integral Test can be correctly applied. The function needs to be continuous over the interval of interest. This assurance means the behavior of the function reflects the series accurately.
When the function is continuous, it becomes feasible to calculate areas under curves (integrals) consistently, ensuring steady navigation through calculus operations like finding limits and rates of change.