2. Use the root test to determine whether the series below converges or diverges. ∞ 2n Σ(1+2n²)n n=1
2. Use the root test to determine whether the series below converges or diverges. ∞ 2n Σ(1+2n²)n n=1
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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Solve the following calc 2 question
![**Exercise 2: Root Test for Convergence**
Determine whether the following series converges or diverges by using the root test:
\[
\sum_{n=1}^{\infty} \frac{n^{2n}}{(1 + 2n^2)^n}
\]
**Explanation:**
This problem requires applying the root test, which is a method used to determine the convergence of an infinite series. The root test involves the following steps:
1. Identify the term \(a_n\) of the series.
2. Calculate \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\).
3. Analyze the limit:
- If the limit is less than 1, the series converges.
- If the limit is greater than 1, the series diverges.
- If the limit equals 1, the test is inconclusive.
The series provided is:
\[
a_n = \frac{n^{2n}}{(1 + 2n^2)^n}
\]
Use the root test to determine the series' behavior.
Work through the steps of calculation to decide if the series converges or diverges, explaining each part in detail as you work through the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33a80c5a-8cc6-4eeb-bbc8-a55f8d96059c%2Ffafacd3f-a199-47ec-b438-4a20f404554d%2F2eqo7twi_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 2: Root Test for Convergence**
Determine whether the following series converges or diverges by using the root test:
\[
\sum_{n=1}^{\infty} \frac{n^{2n}}{(1 + 2n^2)^n}
\]
**Explanation:**
This problem requires applying the root test, which is a method used to determine the convergence of an infinite series. The root test involves the following steps:
1. Identify the term \(a_n\) of the series.
2. Calculate \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\).
3. Analyze the limit:
- If the limit is less than 1, the series converges.
- If the limit is greater than 1, the series diverges.
- If the limit equals 1, the test is inconclusive.
The series provided is:
\[
a_n = \frac{n^{2n}}{(1 + 2n^2)^n}
\]
Use the root test to determine the series' behavior.
Work through the steps of calculation to decide if the series converges or diverges, explaining each part in detail as you work through the problem.
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