2 In(x) -dx%3D 26 2 In(n), does the value of the improper integral tell use about the convergence of the series n6 7=1 the series diverges
2 In(x) -dx%3D 26 2 In(n), does the value of the improper integral tell use about the convergence of the series n6 7=1 the series diverges
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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![**Improper Integral Evaluation and Series Convergence**
Compute the value of the following improper integral, if it converges. (If it diverges, enter ∞ if it diverges to infinity, -∞ if it diverges to negative infinity, or DNE if it diverges for some other reason.) Hint: integrate by parts.
\[
\int_1^{\infty} \frac{2 \ln(x)}{x^6} \, dx = \quad \text{[Enter your answer here]}
\]
**Question:**
What does the value of the improper integral tell us about the convergence of the series \(\sum_{n=1}^{\infty} \frac{2 \ln(n)}{n^6}\)?
- ∘ the series diverges
- ∘ the series converges
- ∘ the Integral Test does not apply](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2F0818ba48-96f4-4ef8-a0e1-785e5c6c0b0d%2Fd9qjqfp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Improper Integral Evaluation and Series Convergence**
Compute the value of the following improper integral, if it converges. (If it diverges, enter ∞ if it diverges to infinity, -∞ if it diverges to negative infinity, or DNE if it diverges for some other reason.) Hint: integrate by parts.
\[
\int_1^{\infty} \frac{2 \ln(x)}{x^6} \, dx = \quad \text{[Enter your answer here]}
\]
**Question:**
What does the value of the improper integral tell us about the convergence of the series \(\sum_{n=1}^{\infty} \frac{2 \ln(n)}{n^6}\)?
- ∘ the series diverges
- ∘ the series converges
- ∘ the Integral Test does not apply
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