Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Σ 4n 2n2 + 1 n = 1 f(x) =
Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Σ 4n 2n2 + 1 n = 1 f(x) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![**Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.**
\[
\sum_{n=1}^{\infty} \frac{4n}{2n^2 + 1}
\]
\[
\int_{1}^{\infty} f(x) = \, \boxed{\phantom{f(x)}}
\]
**Explanation:**
The task is to confirm whether the Integral Test can be applied to the series given by \(\sum_{n=1}^{\infty} \frac{4n}{2n^2 + 1}\), and then use the test to determine if the series converges or diverges.
- **Integral Test Conditions:**
- The function \(f(x) = \frac{4x}{2x^2 + 1}\) must be positive, continuous, and decreasing for \(x \geq 1\).
- **Integral Setup:**
- Calculate the integral \(\int_{1}^{\infty} \frac{4x}{2x^2 + 1} \, dx\) to decide convergence or divergence.
By evaluating whether the integral converges or diverges, conclusions can be drawn about the original series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8160277c-5231-4ec2-b3d7-7a909dd4b420%2F3939a4ac-2d99-4829-9eb6-ad642d6abab8%2Fpd0rpql_processed.png&w=3840&q=75)
Transcribed Image Text:**Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.**
\[
\sum_{n=1}^{\infty} \frac{4n}{2n^2 + 1}
\]
\[
\int_{1}^{\infty} f(x) = \, \boxed{\phantom{f(x)}}
\]
**Explanation:**
The task is to confirm whether the Integral Test can be applied to the series given by \(\sum_{n=1}^{\infty} \frac{4n}{2n^2 + 1}\), and then use the test to determine if the series converges or diverges.
- **Integral Test Conditions:**
- The function \(f(x) = \frac{4x}{2x^2 + 1}\) must be positive, continuous, and decreasing for \(x \geq 1\).
- **Integral Setup:**
- Calculate the integral \(\int_{1}^{\infty} \frac{4x}{2x^2 + 1} \, dx\) to decide convergence or divergence.
By evaluating whether the integral converges or diverges, conclusions can be drawn about the original series.
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