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4. For the series Σ=1+()", does it converge and if so what does it converge to? * n²

4. For the series Σ=1+()", does it converge and if so what does it converge to? * n²

Calculus: Early Transcendentals
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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### Educational Website - Advanced Calculus Problem Set Welcome to the Advanced Calculus Problem Set. Below, you'll find a range of problems designed to challenge your understanding of calculus concepts, including integration, parametric equations, series convergence, and Taylor and Maclaurin series. Each problem encourages the application of theoretical knowledge to practical scenarios. #### Problem Set: 1. **Find the area bounded by \( r = \cos \theta \) for \( -\pi/2 \leq \theta \leq \pi/2 \).** - **Hint:** Use polar coordinates and integration to solve for the area. 2. **Find the length of the \( r(t) = <9t, 4t^3/2, t^2>, 1 \leq t \leq 5 \).** - **Hint:** Compute the magnitude of the derivative of the vector function \( r(t) \). 3. **Find parametric equations of the tangent line to the curve given by \( r(t) = <e^t, t^4, t + 3> \) at the point with \( t = 0 \).** - **Hint:** Determine the derivative of \( r(t) \) and evaluate it at \( t = 0 \). 4. **For the series \( \sum_{m=1}^{\infty} \frac{1}{2^m} \left( \frac{2}{5} \right)^n \), does it converge and if so, what does it converge to?** - **Hint:** Apply the geometric series test to determine convergence. 5. **Determine whether the series \( \sum_{n=1}^{\infty} (-1)^n \cdot \frac{n^2}{n^4+n+1} \) is conditionally convergent, absolutely convergent, or divergent and explain why.** - **Hint:** Use the alternating series test and the comparison test. 6. **Find the radius of convergence and the interval of convergence for \( \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n (x - 2)^n \).** - **Hint:** Use the ratio test to find the radius of convergence. 7. **Find the Taylor Series for \( \cos x \) centered at \( \pi/2 \). What is
Transcribed Image Text:### Educational Website - Advanced Calculus Problem Set Welcome to the Advanced Calculus Problem Set. Below, you'll find a range of problems designed to challenge your understanding of calculus concepts, including integration, parametric equations, series convergence, and Taylor and Maclaurin series. Each problem encourages the application of theoretical knowledge to practical scenarios. #### Problem Set: 1. **Find the area bounded by \( r = \cos \theta \) for \( -\pi/2 \leq \theta \leq \pi/2 \).** - **Hint:** Use polar coordinates and integration to solve for the area. 2. **Find the length of the \( r(t) = <9t, 4t^3/2, t^2>, 1 \leq t \leq 5 \).** - **Hint:** Compute the magnitude of the derivative of the vector function \( r(t) \). 3. **Find parametric equations of the tangent line to the curve given by \( r(t) = <e^t, t^4, t + 3> \) at the point with \( t = 0 \).** - **Hint:** Determine the derivative of \( r(t) \) and evaluate it at \( t = 0 \). 4. **For the series \( \sum_{m=1}^{\infty} \frac{1}{2^m} \left( \frac{2}{5} \right)^n \), does it converge and if so, what does it converge to?** - **Hint:** Apply the geometric series test to determine convergence. 5. **Determine whether the series \( \sum_{n=1}^{\infty} (-1)^n \cdot \frac{n^2}{n^4+n+1} \) is conditionally convergent, absolutely convergent, or divergent and explain why.** - **Hint:** Use the alternating series test and the comparison test. 6. **Find the radius of convergence and the interval of convergence for \( \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n (x - 2)^n \).** - **Hint:** Use the ratio test to find the radius of convergence. 7. **Find the Taylor Series for \( \cos x \) centered at \( \pi/2 \). What is
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