Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 5
Use the Integral Test to determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{1}{(2 n+1)^{3}}$$
Short Answer
Expert verified
The series converges.
Step by step solution
01
Identify the Function
The given series is \( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} \).To apply the integral test, we first identify a continuous function \( f(x) = \frac{1}{(2x+1)^3} \) that approximates the terms of the series as \( x \) becomes large.
02
Check the Conditions
The Integral Test requires that \( f(x) = \frac{1}{(2x+1)^3} \) is positive, continuous, and decreasing for \( x \geq 1 \). Since the denominator \((2x+1)^3\) is always positive for \( x \geq 1 \), \( f(x) \) is positive. \( f(x) \) is continuous because there is no point where the denominator becomes zero. Finally, \( f(x) \) is decreasing because the power \( x^3 \) in the denominator increases much faster than the numerator, thus making the function decrease.
03
Set up the Integral
We need to evaluate the integral \( \int_{1}^{\infty} \frac{1}{(2x+1)^3} \, dx \). This integral will help us determine the convergence of the series.
04
Solve the Integral
To solve the integral \( \int \frac{1}{(2x+1)^3} \, dx \), use a substitution: let \( u = 2x+1 \), then \( du = 2\, dx \) or \( dx = \frac{1}{2}du \). The integral becomes \[ \int \frac{1}{u^3} \cdot \frac{1}{2} \, du = \frac{1}{2} \int u^{-3} \, du. \]The antiderivative of \( u^{-3} \) is \( \frac{u^{-2}}{-2} = -\frac{1}{2u^2} \). Substituting back gives\[-\frac{1}{4(2x+1)^2}.\]
05
Evaluate the Improper Integral
Evaluate the integral from 1 to \( \infty \): \[ \int_{1}^{\infty} \frac{1}{(2x+1)^3} \, dx = \left[ -\frac{1}{4(2x+1)^2} \right]_{1}^{\infty}. \]At \( x = 1 \), this becomes \(-\frac{1}{4 \cdot (3)^2} = -\frac{1}{36}\).As \( x \rightarrow \infty, \ \frac{1}{4(2x+1)^2} \rightarrow 0 \). Hence, the integral evaluates to\[ 0 - \left(-\frac{1}{36}\right) = \frac{1}{36}. \]
06
Conclusion on Convergence
Since the integral \( \int_{1}^{\infty} \frac{1}{(2x+1)^3} \, dx \) is finite and equals \( \frac{1}{36} \), the series \( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} \) converges by the Integral Test.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence and Divergence
When we discuss convergence and divergence in the context of series, we are trying to determine whether an infinite series adds up to a finite number (converges) or grows without bound (diverges). In simpler terms, convergence means the terms of the series get closer to a specific value as more are added.
For example, if each term added to a series makes the total smaller and smaller, ultimately zeroing in on a particular number, the series converges. Conversely, if the sum keeps growing indefinitely, the series diverges.
To solve the problem of whether a series converges or diverges, mathematical tools such as the Integral Test come into play. They help assess the behavior of the series' terms, especially as they approach infinity. Ensuring the underlying function of the series is positive, continuous, and decreasing as required, such tests allow us to use integrals to make conclusions about the series. If the corresponding integral results in a finite number, the series converges; otherwise, it diverges.
For example, if each term added to a series makes the total smaller and smaller, ultimately zeroing in on a particular number, the series converges. Conversely, if the sum keeps growing indefinitely, the series diverges.
To solve the problem of whether a series converges or diverges, mathematical tools such as the Integral Test come into play. They help assess the behavior of the series' terms, especially as they approach infinity. Ensuring the underlying function of the series is positive, continuous, and decreasing as required, such tests allow us to use integrals to make conclusions about the series. If the corresponding integral results in a finite number, the series converges; otherwise, it diverges.
Improper Integrals
Improper integrals are a handy tool in calculus to evaluate integrals where the usual methods don't apply, particularly when dealing with infinite limits or discontinuous functions. They help us investigate functions that could otherwise be challenging to understand, fitting well with the Integral Test for series.
An integral is said to be improper if it has an infinite limit of integration or if the function being integrated has infinite discontinuities within the integration interval. In the context of the Integral Test, we often deal with improper integrals extending to infinity.
Using the proper setup, these integrals let us determine the convergence of a series. As in the solution to the particular series given, evaluating \( \int_{1}^{\infty} \frac{1}{(2x+1)^3} \, dx \) allows us to discern whether the series converges or not. The fact that this integral has a finite value of \( \frac{1}{36} \) means it converges, demonstrating the practical utility of improper integrals in handling such problems.
An integral is said to be improper if it has an infinite limit of integration or if the function being integrated has infinite discontinuities within the integration interval. In the context of the Integral Test, we often deal with improper integrals extending to infinity.
Using the proper setup, these integrals let us determine the convergence of a series. As in the solution to the particular series given, evaluating \( \int_{1}^{\infty} \frac{1}{(2x+1)^3} \, dx \) allows us to discern whether the series converges or not. The fact that this integral has a finite value of \( \frac{1}{36} \) means it converges, demonstrating the practical utility of improper integrals in handling such problems.
Infinite Series
An infinite series is a sum of an infinite sequence of numbers. It extends indefinitely, posing interesting challenges when determining how, or if, it sums to a particular value. While it might seem that adding an endless amount of numbers would always result in a diverging sum, this is not necessarily the case.
Many infinite series converge, meaning they approach a specific value. This typically happens when the terms of the series get smaller and shrink towards zero. But how do we determine if an infinite series converges or diverges? One powerful approach is using convergence tests, such as the Integral Test applied in the original problem.
In assessing the series \( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} \), the function representing the series terms meets the necessary criteria for the integral to be evaluated. When the integral results in a finite value, like \( \frac{1}{36} \), it indicates that the series converges. Understanding infinite series and their behavior is crucial in higher mathematics, particularly in topics like calculus and analysis.
Many infinite series converge, meaning they approach a specific value. This typically happens when the terms of the series get smaller and shrink towards zero. But how do we determine if an infinite series converges or diverges? One powerful approach is using convergence tests, such as the Integral Test applied in the original problem.
In assessing the series \( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} \), the function representing the series terms meets the necessary criteria for the integral to be evaluated. When the integral results in a finite value, like \( \frac{1}{36} \), it indicates that the series converges. Understanding infinite series and their behavior is crucial in higher mathematics, particularly in topics like calculus and analysis.
