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Exam 11: Taylor Polynomials and Infinite Series

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Exam 1: Preliminaries183 Questionsadd course
Exam 2: Functions, Limits, and the Derivative250 Questionsadd course
Exam 3: Differentiation309 Questionsadd course
Exam 4: Applications of the Derivative152 Questionsadd course
Exam 5: Exponential and Logarithmic Functions256 Questionsadd course
Exam 6: Integration291 Questionsadd course
Exam 7: Additional Topics in Integration202 Questionsadd course
Exam 8: Calculus of Several Variables219 Questionsadd course
Exam 9: Differential Equations57 Questionsadd course
Exam 10: Probability and Calculus68 Questionsadd course
Exam 11: Taylor Polynomials and Infinite Series110 Questionsadd course
Exam 12: Trigonometric Functions64 Questionsadd course
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Consider the function Consider the function . ​ Find the Taylor series of the function at the indicated point. ​ __________ ​ Find its radius of convergence. ​ __________ ​ Find the interval of convergence. ​ __________ . ​ Find the Taylor series of the function at the indicated point. ​ __________ ​ Find its radius of convergence. ​ __________ ​ Find the interval of convergence. ​ __________

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Question 61

Find a formula for the capital value of a perpetuity involving payments of P dollars paid at the end of each investment period into a fund that earns interest at the annual rate of k% compounded continuously. ​

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Question 62

Consider the series ​ Consider the series ​ . ​ Find the values of x for which the series converges. ​ Find the sum of the series where it is convergent . ​ Find the values of x for which the series converges. ​ Find the sum of the series where it is convergent

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Question 63

Integrate the power series for Integrate the power series for to obtain a power series representation for the function . to obtain a power series representation for the function Integrate the power series for to obtain a power series representation for the function . .

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Question 64

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​ Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​ Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​ Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​

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Question 65

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​ Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

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Question 66

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 67

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​ Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

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Question 68

Consider the series ​ Consider the series ​ . ​ Determine whether the geometric series converges or diverges. ​ If it converges, find its sum. . ​ Determine whether the geometric series converges or diverges. ​ If it converges, find its sum.

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Question 69

Determine whether the following series is convergent or divergent. ​ Determine whether the following series is convergent or divergent. ​

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Question 70

Find P​4(x), the fourth Taylor polynomial of the following function at the indicated point. ​ Find P​<sub>4</sub>(x), the fourth Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. at Find P​<sub>4</sub>(x), the fourth Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. ​ Express any non-integer coefficients as reduced fractions.

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Question 71

Determine whether the following p-series is convergent or divergent. ​ Determine whether the following p-series is convergent or divergent. ​

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Question 72

Consider the series ​ Consider the series ​ . ​ Determine whether the series converges or diverges. ​ If it converges, find its sum. Round your answer to three decimal places. . ​ Determine whether the series converges or diverges. ​ If it converges, find its sum. Round your answer to three decimal places.

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Question 73

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 74

Find the general term Find the general term of the sequence. ​ of the sequence. ​ Find the general term of the sequence. ​

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Question 75

Write down the first five terms of the sequence. ​ Write down the first five terms of the sequence. ​

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Question 76

Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , between Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , and Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , , Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and ,

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Question 77

Let Let ​ Use the Newton method to find the zero of in the interval . ​ Round your answer to five decimal places, if necessary. ​ Use the Newton method to find the zero of Let ​ Use the Newton method to find the zero of in the interval . ​ Round your answer to five decimal places, if necessary. in the interval Let ​ Use the Newton method to find the zero of in the interval . ​ Round your answer to five decimal places, if necessary. . ​ Round your answer to five decimal places, if necessary.

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Question 78

Differentiate the power series for Differentiate the power series for at to obtain a series representation for the function . at Differentiate the power series for at to obtain a series representation for the function . to obtain a series representation for the function Differentiate the power series for at to obtain a series representation for the function . .

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Question 79

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 80
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