For the next two exercises, use the Root Test to determine if each series converges or diverges. n+1 3 ¹Σ (-m (e² + 1))*** (-1)" 4 n=1 nl+n n=2
For the next two exercises, use the Root Test to determine if each series converges or diverges. n+1 3 ¹Σ (-m (e² + 1))*** (-1)" 4 n=1 nl+n n=2
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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Hello,
Can someone show to solve problem 4 using the root test?
Thank you!
![### Series Convergence Tests
For the next two exercises, use the Ratio Test to determine if each series converges or diverges.
**Exercise 1**
Find the convergence or divergence of the series:
\[ \sum_{n=1}^{\infty} \frac{(-1)^n (n+2)}{3^n} \]
**Exercise 2**
Find the convergence or divergence of the series:
\[ \sum_{n=2}^{\infty} \frac{3n + 2}{\ln n} \]
For the next two exercises, use the Root Test to determine if each series converges or diverges.
**Exercise 3**
Find the convergence or divergence of the series:
\[ \sum_{n=1}^{\infty} \left( -\ln \left( e^2 + \frac{1}{n} \right) \right)^{n+1} \]
**Exercise 4**
Find the convergence or divergence of the series:
\[ \sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1+n}} \]
**Exercise 5**
Determine if the series converges or diverges:
\[ \sum_{n=1}^{\infty} \left( \frac{n-2}{n} \right)^n \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2Fed1ca384-3c8e-4b5c-8a47-593478be8d8e%2Fw3uvcrw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Convergence Tests
For the next two exercises, use the Ratio Test to determine if each series converges or diverges.
**Exercise 1**
Find the convergence or divergence of the series:
\[ \sum_{n=1}^{\infty} \frac{(-1)^n (n+2)}{3^n} \]
**Exercise 2**
Find the convergence or divergence of the series:
\[ \sum_{n=2}^{\infty} \frac{3n + 2}{\ln n} \]
For the next two exercises, use the Root Test to determine if each series converges or diverges.
**Exercise 3**
Find the convergence or divergence of the series:
\[ \sum_{n=1}^{\infty} \left( -\ln \left( e^2 + \frac{1}{n} \right) \right)^{n+1} \]
**Exercise 4**
Find the convergence or divergence of the series:
\[ \sum_{n=2}^{\infty} \frac{(-1)^n}{n^{1+n}} \]
**Exercise 5**
Determine if the series converges or diverges:
\[ \sum_{n=1}^{\infty} \left( \frac{n-2}{n} \right)^n \]
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