Test the series below for convergence using the Root Test. ( 2n +5 3n+3 The limit of the root test simplifies to lim f(n)| where 72-00 f(n) = The limit is: (enter oo for infinity if needed) Based on this, the series
Test the series below for convergence using the Root Test. ( 2n +5 3n+3 The limit of the root test simplifies to lim f(n)| where 72-00 f(n) = The limit is: (enter oo for infinity if needed) Based on this, the series
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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![### Series Convergence Analysis Using the Root Test
#### Problem 1: Root Test for Series Convergence
Test the series below for convergence using the Root Test:
$$\sum_{n=1}^{\infty} \left(\frac{2n + 5}{3n + 3}\right)^n$$
The limit of the root test simplifies to:
$$\lim_{n \to \infty} |f(n)|$$
where
$$f(n) =$$ [Input Required]
The limit is:
$$\lim_{n \to \infty} |f(n)| =$$ [Input Required]
Based on this, the series:
- Ⓐ Converges
- Ⓑ Diverges
---
#### Problem 2: Radius of Convergence of a Series
Find all the values of \( x \) such that the given series would converge:
$$\sum_{n=1}^{\infty} \frac{(-1)^n \cdot 7^n \cdot x^n}{\sqrt{n} + 3}$$
Determine the interval of convergence:
- The series is convergent from \( x = \) [Input Required], with the left end included (enter Y or N): [Input Required]
- The series is convergent to \( x = \) [Input Required], with the right end included (enter Y or N): [Y]
This content aims to help students understand the application of the Root Test and finding the interval of convergence for series in Calculus. Through completing these exercises, students can gain a firmer grasp of determining series behavior and addressing complex calculus problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f2aa9e5-f479-433e-a47a-cf563a6cc086%2F11571e6b-a676-4d21-9ec4-08add912cfdd%2F9q8dtec_processed.png&w=3840&q=75)
Transcribed Image Text:### Series Convergence Analysis Using the Root Test
#### Problem 1: Root Test for Series Convergence
Test the series below for convergence using the Root Test:
$$\sum_{n=1}^{\infty} \left(\frac{2n + 5}{3n + 3}\right)^n$$
The limit of the root test simplifies to:
$$\lim_{n \to \infty} |f(n)|$$
where
$$f(n) =$$ [Input Required]
The limit is:
$$\lim_{n \to \infty} |f(n)| =$$ [Input Required]
Based on this, the series:
- Ⓐ Converges
- Ⓑ Diverges
---
#### Problem 2: Radius of Convergence of a Series
Find all the values of \( x \) such that the given series would converge:
$$\sum_{n=1}^{\infty} \frac{(-1)^n \cdot 7^n \cdot x^n}{\sqrt{n} + 3}$$
Determine the interval of convergence:
- The series is convergent from \( x = \) [Input Required], with the left end included (enter Y or N): [Input Required]
- The series is convergent to \( x = \) [Input Required], with the right end included (enter Y or N): [Y]
This content aims to help students understand the application of the Root Test and finding the interval of convergence for series in Calculus. Through completing these exercises, students can gain a firmer grasp of determining series behavior and addressing complex calculus problems.
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