Determine whether the series converges or diverges. n=1 22nl The series converges by the Limit Comparison Test with a convergent p-series. The series converges by the Direct Comparison Test. Each term is less than that of the harmonic series. O The series diverges by the Direct Comparison Test. Each term is greater than that of a divergent geometri- O The series diverges by the Limit Comparison Test with a divergent geometric series.
Determine whether the series converges or diverges. n=1 22nl The series converges by the Limit Comparison Test with a convergent p-series. The series converges by the Direct Comparison Test. Each term is less than that of the harmonic series. O The series diverges by the Direct Comparison Test. Each term is greater than that of a divergent geometri- O The series diverges by the Limit Comparison Test with a divergent geometric 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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![### Determining Convergence or Divergence of a Series
#### Problem Statement:
Determine whether the following series converges or diverges:
\[ \sum_{n=1}^{\infty} \frac{22n!}{n^n} \]
#### Multiple Choice Options:
1. **The series converges by the Limit Comparison Test with a convergent \( p \)-series.**
2. The series converges by the Direct Comparison Test. Each term is less than that of the harmonic series.
3. The series diverges by the Direct Comparison Test. Each term is greater than that of a divergent geometric series.
4. The series diverges by the Limit Comparison Test with a divergent geometric series.
#### Explanation:
1. **Limit Comparison Test with a \( p \)-series:**
This option involves comparing the given series to a known \( p \)-series:
\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]
If \( p > 1 \), the \( p \)-series converges; otherwise, it diverges. The Limit Comparison Test involves checking whether the limit
\[ \lim_{n \to \infty} \frac{a_n}{b_n} \]
exists and is finite and positive, where \( a_n \) is the term of the series we are investigating and \( b_n \) is the term of the known \( p \)-series.
2. **Direct Comparison Test with the Harmonic Series:**
The harmonic series is given by:
\[ \sum_{n=1}^{\infty} \frac{1}{n} \]
If each term of the given series is less than the corresponding term of the harmonic series, and the harmonic series is known to diverge, this comparison does not by itself guarantee convergence.
3. **Direct Comparison Test with a Divergent Geometric Series:**
A divergent geometric series can be of the form:
\[ \sum_{n=0}^{\infty} ar^n \]
where \( |r| \geq 1 \). If the terms of the given series are greater than those of a divergent geometric series, then the given series diverges.
4. **Limit Comparison Test with a Divergent Geometric Series:**
Similar to option 1, but the comparison is made with a known divergent geometric series](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18d7602f-9f25-445c-bda8-4d38e99e2d82%2Fee6227d8-4c15-4b87-b6c8-e08a3a16f31a%2Fvrfgh8q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining Convergence or Divergence of a Series
#### Problem Statement:
Determine whether the following series converges or diverges:
\[ \sum_{n=1}^{\infty} \frac{22n!}{n^n} \]
#### Multiple Choice Options:
1. **The series converges by the Limit Comparison Test with a convergent \( p \)-series.**
2. The series converges by the Direct Comparison Test. Each term is less than that of the harmonic series.
3. The series diverges by the Direct Comparison Test. Each term is greater than that of a divergent geometric series.
4. The series diverges by the Limit Comparison Test with a divergent geometric series.
#### Explanation:
1. **Limit Comparison Test with a \( p \)-series:**
This option involves comparing the given series to a known \( p \)-series:
\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]
If \( p > 1 \), the \( p \)-series converges; otherwise, it diverges. The Limit Comparison Test involves checking whether the limit
\[ \lim_{n \to \infty} \frac{a_n}{b_n} \]
exists and is finite and positive, where \( a_n \) is the term of the series we are investigating and \( b_n \) is the term of the known \( p \)-series.
2. **Direct Comparison Test with the Harmonic Series:**
The harmonic series is given by:
\[ \sum_{n=1}^{\infty} \frac{1}{n} \]
If each term of the given series is less than the corresponding term of the harmonic series, and the harmonic series is known to diverge, this comparison does not by itself guarantee convergence.
3. **Direct Comparison Test with a Divergent Geometric Series:**
A divergent geometric series can be of the form:
\[ \sum_{n=0}^{\infty} ar^n \]
where \( |r| \geq 1 \). If the terms of the given series are greater than those of a divergent geometric series, then the given series diverges.
4. **Limit Comparison Test with a Divergent Geometric Series:**
Similar to option 1, but the comparison is made with a known divergent geometric series
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