Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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calculus medium, please solve questionnnn 5
![### Calculus and Series Problems
#### Problem 1
**The series:**
\[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^4/5} \]
**Options:**
- A) Conditionally convergent
- B) Absolutely convergent
- C) Divergent
- D) D.N.E
#### Problem 2
**The sequence:**
\[ \left\{ (1 - \frac{7}{n})^n \right\}_{n=1}^{\infty} \]
**Options:**
- A) Converges to \( e^7 \)
- B) Converges to \( e^{-7} \)
- C) Converges to \(-7\)
- D) Diverges
#### Problem 3
\[ \int_{3}^{\infty} \frac{4}{x^2 + 49} \, dx = ? \]
**Options:**
- A) \( \frac{\pi}{7} \)
- B) \( \frac{\pi}{2} \)
- C) \( \frac{\pi}{3} \)
- D) Diverges
#### Problem 4
**The interval of convergence of the power series:**
\[ \sum_{k=1}^{\infty} \left( \frac{x}{3} \right)^k \]
**Options:**
- A) [0]
- B) \(\left(-\frac{1}{3}, \frac{1}{3} \right)\)
- C) \((-3, 3)\)
- D) \((-∞, ∞)\)
#### Problem 5
**The appropriate substitution for:**
\[ \int \frac{dx}{\sqrt{x^2 + 2x + 10}} \]
**Options:**
- A) \( x + 1 = 3 \tan \theta \)
- B) \( x = 10 \sin \theta \)
- C) \( x - 1 = 2 \tan \theta \)
- D) \( x - 1 = 3 \tan \theta \)
#### Problem 6
\[ \int \frac{x}{(x+2)^2} \, dx = ? \]
**Options:**
- A) \( \frac{x}{x+3} + \ln|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5231f5b4-e859-48c1-839f-bbb3459ab86b%2F637a986a-ffd1-4d7e-82fe-58a2ae722142%2Fct905j9_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus and Series Problems
#### Problem 1
**The series:**
\[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^4/5} \]
**Options:**
- A) Conditionally convergent
- B) Absolutely convergent
- C) Divergent
- D) D.N.E
#### Problem 2
**The sequence:**
\[ \left\{ (1 - \frac{7}{n})^n \right\}_{n=1}^{\infty} \]
**Options:**
- A) Converges to \( e^7 \)
- B) Converges to \( e^{-7} \)
- C) Converges to \(-7\)
- D) Diverges
#### Problem 3
\[ \int_{3}^{\infty} \frac{4}{x^2 + 49} \, dx = ? \]
**Options:**
- A) \( \frac{\pi}{7} \)
- B) \( \frac{\pi}{2} \)
- C) \( \frac{\pi}{3} \)
- D) Diverges
#### Problem 4
**The interval of convergence of the power series:**
\[ \sum_{k=1}^{\infty} \left( \frac{x}{3} \right)^k \]
**Options:**
- A) [0]
- B) \(\left(-\frac{1}{3}, \frac{1}{3} \right)\)
- C) \((-3, 3)\)
- D) \((-∞, ∞)\)
#### Problem 5
**The appropriate substitution for:**
\[ \int \frac{dx}{\sqrt{x^2 + 2x + 10}} \]
**Options:**
- A) \( x + 1 = 3 \tan \theta \)
- B) \( x = 10 \sin \theta \)
- C) \( x - 1 = 2 \tan \theta \)
- D) \( x - 1 = 3 \tan \theta \)
#### Problem 6
\[ \int \frac{x}{(x+2)^2} \, dx = ? \]
**Options:**
- A) \( \frac{x}{x+3} + \ln|
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