Which of the following statements is false? Select all that apply. If 1000 terms are added to a convergent series, then it is possible that the new series is divergent. The Divergence Test can never be used to conclude that a series converges. Let an then f(n), where f(x) is a continuous, positive, decreasing function for all x ≥ 1. If Σa₁ - 100 an = n=1 [ f(x) dx. If f f(x) The integral test can be applied to divergent series. dx converges,
Which of the following statements is false? Select all that apply. If 1000 terms are added to a convergent series, then it is possible that the new series is divergent. The Divergence Test can never be used to conclude that a series converges. Let an then f(n), where f(x) is a continuous, positive, decreasing function for all x ≥ 1. If Σa₁ - 100 an = n=1 [ f(x) dx. If f f(x) The integral test can be applied to divergent series. dx converges,
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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![### Educational Website Content
#### Question: Identifying False Statements in Series Analysis
**Which of the following statements is false? Select all that apply.**
1. **If 1000 terms are added to a convergent series, then it is possible that the new series is divergent.**
- This statement is checked.
2. **The Divergence Test can never be used to conclude that a series converges.**
- This statement is also checked.
3. Let \( a_n = f(n) \), where \( f(x) \) is a continuous, positive, decreasing function for all \( x \geq 1 \). If \( \int_{1}^{\infty} f(x) \, dx \) converges, then
\[
\sum_{n=1}^{\infty} a_n = \int_{1}^{\infty} f(x) \, dx.
\]
- This statement is not checked.
4. **The integral test can be applied to divergent series.**
- This statement is not checked.
**Explanation:**
- **Statement 1:** Checked as false. Adding a finite number of terms to a convergent series cannot change its convergence nature, it will still converge.
- **Statement 2:** Checked as true. The Divergence Test (or nth-term test for divergence) states that if \( \lim_{n \to \infty} a_n \neq 0 \) or does not exist, then the series \( \sum a_n \) diverges. However, if \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive about convergence.
- **Statement 3:** This is the Integral Test, which relates the convergence of a series to the convergence of an improper integral. If the integral converges, so does the series, and vice versa.
- **Statement 4:** The integral test applies to both convergent and divergent series, depending on the behavior of the integral. Thus, this statement should be checked as false if determining if it could exclusively apply.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42d7a143-2adc-420d-86c3-225c6cc98b5a%2F1850cadf-6fef-4848-b59d-3375797b0c73%2Fxfhdq3n_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
#### Question: Identifying False Statements in Series Analysis
**Which of the following statements is false? Select all that apply.**
1. **If 1000 terms are added to a convergent series, then it is possible that the new series is divergent.**
- This statement is checked.
2. **The Divergence Test can never be used to conclude that a series converges.**
- This statement is also checked.
3. Let \( a_n = f(n) \), where \( f(x) \) is a continuous, positive, decreasing function for all \( x \geq 1 \). If \( \int_{1}^{\infty} f(x) \, dx \) converges, then
\[
\sum_{n=1}^{\infty} a_n = \int_{1}^{\infty} f(x) \, dx.
\]
- This statement is not checked.
4. **The integral test can be applied to divergent series.**
- This statement is not checked.
**Explanation:**
- **Statement 1:** Checked as false. Adding a finite number of terms to a convergent series cannot change its convergence nature, it will still converge.
- **Statement 2:** Checked as true. The Divergence Test (or nth-term test for divergence) states that if \( \lim_{n \to \infty} a_n \neq 0 \) or does not exist, then the series \( \sum a_n \) diverges. However, if \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive about convergence.
- **Statement 3:** This is the Integral Test, which relates the convergence of a series to the convergence of an improper integral. If the integral converges, so does the series, and vice versa.
- **Statement 4:** The integral test applies to both convergent and divergent series, depending on the behavior of the integral. Thus, this statement should be checked as false if determining if it could exclusively apply.
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