Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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## Calculus Problems and Solutions ### Problem 005 **Find the sum of the series** \[ \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n}}{n!} \] Options: 1. \( \text{sum} = e^{-2} \) 2. \( \text{sum} = 4 \) 3. \( \text{sum} = e^{2} \) 4. \( \text{sum} = e^{-4} \) 5. \( \text{sum} = e^{4} \) 6. \( \text{sum} = 2 \) ### Problem 006 **Use an appropriate Taylor series to evaluate** \[ \lim_{x \to 0} \frac{\sin(3x) - 3x + 2x^3}{x^3} \] Options: 1. \( \text{limit} = -\frac{11}{6} \) 2. \( \text{limit} = -\frac{5}{2} \) 3. \( \text{limit} = -\frac{13}{6} \) 4. \( \text{limit} = -\frac{19}{6} \) 5. \( \text{limit} = -\frac{17}{6} \) ### Problem 007 **Use an appropriate Taylor series representation to evaluate** \[ \lim_{x \to 0} \frac{3x - \tan^{-1}(3x)}{6x^3} \] Options: 1. \( \text{limit} = \frac{3}{4} \) 2. \( \text{limit} = \frac{5}{4} \) 3. \( \text{limit} = \frac{3}{2} \) 4. \( \text{limit} = \frac{9}{2} \) 5. \( \text{limit} = \frac{5}{2} \) ### Problem 008 **Find a power series representation for the function** \[ f(x) = 2^x \] **on** \( (-\infty, \infty) \). Options: 1. \( f(x) = \sum_{n=0}^{\infty} \frac{(\ln(2))^n}{n
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Transcribed Image Text:## Calculus Problems and Solutions ### Problem 005 **Find the sum of the series** \[ \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n}}{n!} \] Options: 1. \( \text{sum} = e^{-2} \) 2. \( \text{sum} = 4 \) 3. \( \text{sum} = e^{2} \) 4. \( \text{sum} = e^{-4} \) 5. \( \text{sum} = e^{4} \) 6. \( \text{sum} = 2 \) ### Problem 006 **Use an appropriate Taylor series to evaluate** \[ \lim_{x \to 0} \frac{\sin(3x) - 3x + 2x^3}{x^3} \] Options: 1. \( \text{limit} = -\frac{11}{6} \) 2. \( \text{limit} = -\frac{5}{2} \) 3. \( \text{limit} = -\frac{13}{6} \) 4. \( \text{limit} = -\frac{19}{6} \) 5. \( \text{limit} = -\frac{17}{6} \) ### Problem 007 **Use an appropriate Taylor series representation to evaluate** \[ \lim_{x \to 0} \frac{3x - \tan^{-1}(3x)}{6x^3} \] Options: 1. \( \text{limit} = \frac{3}{4} \) 2. \( \text{limit} = \frac{5}{4} \) 3. \( \text{limit} = \frac{3}{2} \) 4. \( \text{limit} = \frac{9}{2} \) 5. \( \text{limit} = \frac{5}{2} \) ### Problem 008 **Find a power series representation for the function** \[ f(x) = 2^x \] **on** \( (-\infty, \infty) \). Options: 1. \( f(x) = \sum_{n=0}^{\infty} \frac{(\ln(2))^n}{n
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