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Magazine Chapter 10: Problem 17
Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied. $$\sum_{k=0}^{\infty} \frac{1}{\sqrt{k+8}}$$
Short Answer
Expert verified
Based on the conditions tested and the Integral Test, the series $\sum_{k=0}^{\infty} \frac{1}{\sqrt{k+8}}$ is found to diverge.
Step by step solution
01
Define the function
First, let's define the function \(f(x)\) that represents the terms of the series as follows:
$$f(x) = \frac{1}{\sqrt{x+8}}$$
02
Check Continuity, Positivity, and Being Decreasing
Now, we need to check if the function \(f(x)\) is continuous, positive, and decreasing for \(x \geq 0\).
1. Continuity: The function is a rational function where the denominator is never zero for \(x \geq 0\), so it's continuous for all \(x \geq 0\).
2. Positivity: Since the function involves a square root in the denominator and \(x+8 \geq 8\) for \(x \geq 0\), it's always positive for \(x \geq 0\).
3. Decreasing: To check if the function is decreasing, we can find its derivative and check if it's negative for all \(x \geq 0\). The derivative of \(f(x)\) is:
$$f'(x) = \frac{-1}{2\sqrt{(x+8)^3}}$$
Considering that \(x+8 \geq 8\) for \(x \geq 0\), this implies that the derivative is negative for all \(x \geq 0\). Thus, the function is decreasing for \(x \geq 0\).
Since \(f(x)\) is continuous, positive, and decreasing for \(x \geq 0\), we can now apply the Integral Test.
03
Apply the Integral Test
The Integral Test states that if the sum of the series \(f(n)\) converges, then the integral of the function \(f(x)\) also converges. To apply the Integral Test, we need to find the integral of the function \(f(x)\) from \(0\) to \(\infty\):
$$\int_{0}^{\infty} \frac{1}{\sqrt{x+8}} dx $$
To solve the integral, we can use the substitution method, let \(u=x+8\), then \(du=dx\). The integral becomes:
$$\int_{0+8}^{\infty} \frac{1}{\sqrt{u}} du $$
Now, we can integrate and evaluate the integral:
$$\left[2\sqrt{u} \right]_{8}^{\infty}=2(\sqrt{\infty}-\sqrt{8})$$
Since the integral is infinite, it diverges.
04
Conclusion
According to the Integral Test, since the integral of the function corresponding to the given series diverges, the series also diverges. Therefore, the series $$\sum_{k=0}^{\infty} \frac{1}{\sqrt{k+8}}$$ diverges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence of Series
Convergence of series is a fundamental concept in mathematical analysis, particularly in the context of infinite series. An infinite series is said to converge if the sum of its terms approaches a specific value as the number of terms increases indefinitely. Conversely, a series diverges if the sum does not settle on a particular value but instead continues to increase, decrease, or vary without approaching a finite limit.
The decision to declare a series as convergent or divergent directly impacts mathematical and physical interpretations, as convergence often indicates a certain kind of stability or predictability within the series. For example, the convergence of a power series is the cornerstone of Taylor series and Fourier series that are widely used in physics and engineering to represent complex functions more simply.
Various tests exist for determining the convergence of series, such as the Comparison Test, Ratio Test, and Root Test. In the given exercise, the Integral Test, a powerful tool for this purpose, was correctly applied. The Integral Test compares the series to an improper integral, stating that if the integral of the function that defines the series terms converges, then the series converges, and if the integral diverges, so does the series. This test requires the function to be continuous, positive, and decreasing on the interval of summation.
The decision to declare a series as convergent or divergent directly impacts mathematical and physical interpretations, as convergence often indicates a certain kind of stability or predictability within the series. For example, the convergence of a power series is the cornerstone of Taylor series and Fourier series that are widely used in physics and engineering to represent complex functions more simply.
Various tests exist for determining the convergence of series, such as the Comparison Test, Ratio Test, and Root Test. In the given exercise, the Integral Test, a powerful tool for this purpose, was correctly applied. The Integral Test compares the series to an improper integral, stating that if the integral of the function that defines the series terms converges, then the series converges, and if the integral diverges, so does the series. This test requires the function to be continuous, positive, and decreasing on the interval of summation.
Infinite Series
An infinite series is an expression created by adding an infinite sequence of terms together. These series are ubiquitous in different branches of mathematics, from calculus to complex analysis. When dealing with infinite series, it's not just about the addition of numbers but understanding the behavior of the sum as more and more terms are included.
To reveal the nature of infinite series, mathematicians evaluate whether the series converges or diverges. Convergence is achieved when the sum approaches a finite limit as the number of terms grows, an essential property for series that can represent physical quantities or mathematical constants. A famous example of a convergent infinite series is the geometric series, which has vast applications in computing and engineering scenarios.
In the context of the problem provided, the infinite series in question is a harmonic type, characterized by the reciprocal of a square root function. Although each term decreases in size, depending on their rate of decay, the overall sum may not necessarily converge to a limit, as demonstrated by the Integral Test carried out in the solution.
To reveal the nature of infinite series, mathematicians evaluate whether the series converges or diverges. Convergence is achieved when the sum approaches a finite limit as the number of terms grows, an essential property for series that can represent physical quantities or mathematical constants. A famous example of a convergent infinite series is the geometric series, which has vast applications in computing and engineering scenarios.
In the context of the problem provided, the infinite series in question is a harmonic type, characterized by the reciprocal of a square root function. Although each term decreases in size, depending on their rate of decay, the overall sum may not necessarily converge to a limit, as demonstrated by the Integral Test carried out in the solution.
Improper Integrals
Improper integrals are integrals that deal with unbounded intervals or discontinuous integrands at certain points within the interval of integration. They are considered 'improper' because traditional methods of integration assume bounded intervals and continuous functions. An improper integral can have infinite limits of integration, such as from a finite number to infinity, or the integrand can behave in an unbounded manner within the interval.
To evaluate an improper integral, we often turn to limit processes, where the integral is taken over a bounded region, and then the bound is extended toward infinity, or a point of discontinuity is approached as a limit. If the resulting limit exists and is finite, the improper integral is said to converge; otherwise, it diverges.
In the textbook solution, the integral used in the Integral Test is an example of an improper integral: the limit of integration extends to infinity. As shown by the test, the failure of the improper integral to converge (it diverges to infinity) implies that the corresponding infinite series also diverges. This is a direct demonstration of how the behavior of improper integrals can inform the understanding of infinite series and their convergence properties.
To evaluate an improper integral, we often turn to limit processes, where the integral is taken over a bounded region, and then the bound is extended toward infinity, or a point of discontinuity is approached as a limit. If the resulting limit exists and is finite, the improper integral is said to converge; otherwise, it diverges.
In the textbook solution, the integral used in the Integral Test is an example of an improper integral: the limit of integration extends to infinity. As shown by the test, the failure of the improper integral to converge (it diverges to infinity) implies that the corresponding infinite series also diverges. This is a direct demonstration of how the behavior of improper integrals can inform the understanding of infinite series and their convergence properties.
