Question

Use the Integral Test to determine the convergence of the given series. $$\sum_{n=1}^{\infty} \frac{1}{n^{4}}$$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{1}{n^4} \) converges.

Step-by-step solution

Identify the Function

To use the Integral Test, we identify a function that models the terms of the series. In this case, the function is \( f(x) = \frac{1}{x^4} \). This is a positive, continuous, and decreasing function for \( x \geq 1 \).

Set Up the Integral

We set up the improper integral \( \int_{1}^{\infty} \frac{1}{x^4} \, dx \) to evaluate and compare with the series. The convergence of this integral will help determine the convergence of the series.

Evaluate the Integral

First, find the antiderivative: \( \int \frac{1}{x^4} \, dx = -\frac{1}{3x^3} + C \). Now evaluate the improper integral from 1 to \( \infty \):\[ \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^4} \, dx = \lim_{b \to \infty} \left(-\frac{1}{3b^3} - \left(-\frac{1}{3(1)^3}\right)\right). \]

Simplify the Expression

Simplify the expression using the limits: \( \lim_{b \to \infty} \left(-\frac{1}{3b^3} + \frac{1}{3}\right) = \frac{1}{3} \). As \( b \to \infty \), \(-\frac{1}{3b^3} \to 0\), leaving us with \( \frac{1}{3} \).

Conclusion on Convergence

Since the improper integral converges to \( \frac{1}{3} \), the Integral Test tells us that the series \( \sum_{n=1}^{\infty} \frac{1}{n^4} \) also converges.

Key concepts

Convergence of Series

Understanding the convergence of a series is crucial in determining whether the sum of its infinite terms results in a finite value. In the exercise provided, we are tasked to determine the convergence of the series \( \sum_{n=1}^{\infty} \frac{1}{n^{4}} \).

The Integral Test is a powerful tool in calculus used to establish the convergence or divergence of an infinite series. It applies specifically to series whose terms come from positive, continuous, and decreasing functions. By analyzing the behavior of an integral instead of the series directly, we can conclude about the series’ convergence.

• If the corresponding improper integral converges, the series converges.
• If the integral diverges, the series diverges as well.

This method provides a comparative and sometimes an easier evaluation path, particularly when straightforward sum calculations are challenging.

Improper Integral

An improper integral is an integral that requires one or more of the limits of integration to be infinite or where the integrand becomes infinite within the limits of integration. In our case, the improper integral \( \int_{1}^{\infty} \frac{1}{x^4} \, dx \) assesses whether the function \( f(x) = \frac{1}{x^4} \) summed over an infinite interval can yield a finite area under the curve.

To handle the infinite upper limit, we often evaluate this type of integral using a limit:

• First, define the integral from a to b, \( \int_{a}^{b} f(x) \, dx \).
• Then, evaluate the limit of this definite integral as b approaches infinity, \( \lim_{b \to \infty} \int_{a}^{b} f(x) \, dx \).

This process converts the issue from an infinite sum into a finite evaluation followed by limit considerations. If the result is finite, the improper integral converges, implying the related series does as well.

Antiderivative

Finding the antiderivative is a critical step in computing improper integrals, as it allows us to transition from integration to actual valuation. The antiderivative of a function, sometimes referred to as the indefinite integral, is another function whose derivative gives back the original function.

In the exercise, the function \( f(x) = \frac{1}{x^4} \) needed its antiderivative to progress in evaluating the integral. The antiderivative of \( \frac{1}{x^4} \) is \( -\frac{1}{3x^3} + C \). This expression helps us to evaluate the definite integral\[ \int \frac{1}{x^4} \, dx = -\frac{1}{3x^3} + C \]From here, we use this result in calculating the improper integral from 1 to \( \infty \), a vital conversion to solve the problem. By simplifying further with limits, we determine the finite nature of our infinite boundary.

Infinite Series

An infinite series is a sum of infinitely many terms, typically expressed in the form \( \sum_{n=1}^{\infty} a_n \). The key question usually revolves around whether this infinite summation converges to a finite value. If a series converges, irrespective of its infinite nature, it accumulates to a certain limit, making it manageable for mathematical analysis.

In our specific problem, the series \( \sum_{n=1}^{\infty} \frac{1}{n^4} \) is an infinite series integral to many considerations such as in physics, engineering, and other science fields.

• Convergence makes an infinite series useful and applicable in real-world scenarios.
• If a series diverges, its partial sums grow without bound or do not approach a specific value.

Integrating series concepts with methods like the Integral Test bridges infinite mathematical theory to practical convergence conclusions.