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**Mathematics Multiple Choice Questions** 1. **Suppose that \( x = 3 \ln y \).** 2. **The coefficient of \( x^3 \) in the Maclaurin series expansion of \( e^{2x} \) is:** - A) \( 8 \) - B) \( 2 \) - C) \( \frac{2}{3} \) - D) \( \frac{4}{3} \) 3. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( \frac{1}{3} \) - B) \( \frac{1}{6} \) - C) \( \frac{1}{2} \) - D) \( \frac{2}{9} \) 4. **For what values of \( p \) will both series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) and \( \sum_{k=1}^{\infty} \frac{1}{k^{2p}} \) converge?** - A) \( \frac{1}{2} < p < 2 \) - B) \( -\frac{1}{2} < p < 1 \) - C) \( p < \frac{1}{2} \) and \( p > 2 \) - D) \( \frac{1}{2} \leq p < 2 \) 5. **According to the method of partial fractions, \( \frac{x}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3} \), the value of \( B \) is:** - A) \( -2 \) - B) \( \frac{1}{2} \) - C) \( -1 \) - D) \( 3 \) 6. **Which of the following series converge:** - A) \( \sum_{k=0}^{\infty} \frac{1}{k

**Mathematics Multiple Choice Questions** 1. **Suppose that \( x = 3 \ln y \).** 2. **The coefficient of \( x^3 \) in the Maclaurin series expansion of \( e^{2x} \) is:** - A) \( 8 \) - B) \( 2 \) - C) \( \frac{2}{3} \) - D) \( \frac{4}{3} \) 3. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( \frac{1}{3} \) - B) \( \frac{1}{6} \) - C) \( \frac{1}{2} \) - D) \( \frac{2}{9} \) 4. **For what values of \( p \) will both series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) and \( \sum_{k=1}^{\infty} \frac{1}{k^{2p}} \) converge?** - A) \( \frac{1}{2} < p < 2 \) - B) \( -\frac{1}{2} < p < 1 \) - C) \( p < \frac{1}{2} \) and \( p > 2 \) - D) \( \frac{1}{2} \leq p < 2 \) 5. **According to the method of partial fractions, \( \frac{x}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3} \), the value of \( B \) is:** - A) \( -2 \) - B) \( \frac{1}{2} \) - C) \( -1 \) - D) \( 3 \) 6. **Which of the following series converge:** - A) \( \sum_{k=0}^{\infty} \frac{1}{k

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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calculus medium, please solve questionnnn 17

**Mathematics Multiple Choice Questions** 1. **Suppose that \( x = 3 \ln y \).** 2. **The coefficient of \( x^3 \) in the Maclaurin series expansion of \( e^{2x} \) is:** - A) \( 8 \) - B) \( 2 \) - C) \( \frac{2}{3} \) - D) \( \frac{4}{3} \) 3. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( \frac{1}{3} \) - B) \( \frac{1}{6} \) - C) \( \frac{1}{2} \) - D) \( \frac{2}{9} \) 4. **For what values of \( p \) will both series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) and \( \sum_{k=1}^{\infty} \frac{1}{k^{2p}} \) converge?** - A) \( \frac{1}{2} < p < 2 \) - B) \( -\frac{1}{2} < p < 1 \) - C) \( p < \frac{1}{2} \) and \( p > 2 \) - D) \( \frac{1}{2} \leq p < 2 \) 5. **According to the method of partial fractions, \( \frac{x}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3} \), the value of \( B \) is:** - A) \( -2 \) - B) \( \frac{1}{2} \) - C) \( -1 \) - D) \( 3 \) 6. **Which of the following series converge:** - A) \( \sum_{k=0}^{\infty} \frac{1}{k
Transcribed Image Text:**Mathematics Multiple Choice Questions** 1. **Suppose that \( x = 3 \ln y \).** 2. **The coefficient of \( x^3 \) in the Maclaurin series expansion of \( e^{2x} \) is:** - A) \( 8 \) - B) \( 2 \) - C) \( \frac{2}{3} \) - D) \( \frac{4}{3} \) 3. **In the Taylor series generated by \( f(x) = x^{1/3} \) and \( x_0 = 1 \), the coefficient of \( (x-1)^2 \) is:** - A) \( \frac{1}{3} \) - B) \( \frac{1}{6} \) - C) \( \frac{1}{2} \) - D) \( \frac{2}{9} \) 4. **For what values of \( p \) will both series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) and \( \sum_{k=1}^{\infty} \frac{1}{k^{2p}} \) converge?** - A) \( \frac{1}{2} < p < 2 \) - B) \( -\frac{1}{2} < p < 1 \) - C) \( p < \frac{1}{2} \) and \( p > 2 \) - D) \( \frac{1}{2} \leq p < 2 \) 5. **According to the method of partial fractions, \( \frac{x}{(x-1)(x-2)(x-3)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3} \), the value of \( B \) is:** - A) \( -2 \) - B) \( \frac{1}{2} \) - C) \( -1 \) - D) \( 3 \) 6. **Which of the following series converge:** - A) \( \sum_{k=0}^{\infty} \frac{1}{k
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