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4" (18) The coefficient of x³ in the Maclaurin series expansion of exis: A) - B) D) 3 (19) The series (-1) 4n

4" (18) The coefficient of x³ in the Maclaurin series expansion of exis: A) - B) D) 3 (19) The series (-1) 4n

Advanced Engineering Mathematics
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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please solve questionnnn 18

# Calculus and Series Practice Problems ## Problem Set 1. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( 1 \) - B) \( 1/4 \) - C) \( -1/4 \) - D) \( 2/9 \) 2. **The integral \( \int e^{3x} dx \) is:** - A) \( 1/3 e^{3x} + c \) - B) \( 2 + \ln|x + 2| + c \) - C) \( -1 / (x + 2) + c \) - D) \( \ln|x + 2| + c \) 3. **The sum \( \sum_{n=1}^{\infty} \frac{1}{n+1} - \sum_{n=1}^{\infty} \frac{1}{n+2} = \):** - A) \( 1/3 \) - B) \( 3/2 \) - C) \( 2 \) - D) \( 3/2 \) 4. **The integral \( \int \frac{dx}{x^3 + x} \) is:** - A) \( -1/2 \ln(x) + c \) - B) \( \sec^{-1}(\sin \theta) + c \) - C) \( \ln \left( \frac{x - \sin \theta}{2 \sin \theta} \right) + c \) - D) \( \tan^{-1}(\sin \theta) + c \) 5. **Which of the following series converge?** - A) \( \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \) - B) \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) - C) \( \sum_{k=1}^{\infty} \frac{1}{k} \) - D) \( \sum_{k=m
Transcribed Image Text:# Calculus and Series Practice Problems ## Problem Set 1. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( 1 \) - B) \( 1/4 \) - C) \( -1/4 \) - D) \( 2/9 \) 2. **The integral \( \int e^{3x} dx \) is:** - A) \( 1/3 e^{3x} + c \) - B) \( 2 + \ln|x + 2| + c \) - C) \( -1 / (x + 2) + c \) - D) \( \ln|x + 2| + c \) 3. **The sum \( \sum_{n=1}^{\infty} \frac{1}{n+1} - \sum_{n=1}^{\infty} \frac{1}{n+2} = \):** - A) \( 1/3 \) - B) \( 3/2 \) - C) \( 2 \) - D) \( 3/2 \) 4. **The integral \( \int \frac{dx}{x^3 + x} \) is:** - A) \( -1/2 \ln(x) + c \) - B) \( \sec^{-1}(\sin \theta) + c \) - C) \( \ln \left( \frac{x - \sin \theta}{2 \sin \theta} \right) + c \) - D) \( \tan^{-1}(\sin \theta) + c \) 5. **Which of the following series converge?** - A) \( \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \) - B) \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) - C) \( \sum_{k=1}^{\infty} \frac{1}{k} \) - D) \( \sum_{k=m
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