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4. In this problem we'll figure out how to represent the function h(x) = ln(1-x) as a power series, (a) Find the derivative of h(r). (Remember there is a little chain rule involved here.) h'(a) = (b) We can recognize h'(x) as similar to the expression we see in the geometric series formula. What is a? What is r? Use this to represent h'(x) as a power series. -1 T=\ -X h'(x) = (c) Find the antiderivative of your power series (term by term) to obtain a power series representation of h(r) itself. h(x) = ln(1-x) = (d) Plug in x = 0 to solve for the value of C. + C (e) Using Desmos or your graphing calculator, graph h(r). Also graph that power series partial sum that includes the first six terms. Are your graphs close to each other? For what interval of r-values is this a good approximation? (f) Expand your power series partial sum to include more terms. What do you notice on the graph? (g) What is the interval of convergence of your power series? What is the radius of convergence?

4. In this problem we'll figure out how to represent the function h(x) = ln(1-x) as a power series, (a) Find the derivative of h(r). (Remember there is a little chain rule involved here.) h'(a) = (b) We can recognize h'(x) as similar to the expression we see in the geometric series formula. What is a? What is r? Use this to represent h'(x) as a power series. -1 T=\ -X h'(x) = (c) Find the antiderivative of your power series (term by term) to obtain a power series representation of h(r) itself. h(x) = ln(1-x) = (d) Plug in x = 0 to solve for the value of C. + C (e) Using Desmos or your graphing calculator, graph h(r). Also graph that power series partial sum that includes the first six terms. Are your graphs close to each other? For what interval of r-values is this a good approximation? (f) Expand your power series partial sum to include more terms. What do you notice on the graph? (g) What is the interval of convergence of your power series? What is the radius of convergence?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 14EQ
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4. In this problem we'll figure out how to represent the function h(x) = ln(1-x) as a power series, (a) Find the derivative of h(r). (Remember there is a little chain rule involved here.) h'(a) = (b) We can recognize h'(x) as similar to the expression we see in the geometric series formula. What is a? What is r? Use this to represent h'(x) as a power series. -1 T=\ -X h'(x) = (c) Find the antiderivative of your power series (term by term) to obtain a power series representation of h(r) itself. h(x) = ln(1-x) = (d) Plug in x = 0 to solve for the value of C. + C (e) Using Desmos or your graphing calculator, graph h(r). Also graph that power series partial sum that includes the first six terms. Are your graphs close to each other? For what interval of r-values is this a good approximation? (f) Expand your power series partial sum to include more terms. What do you notice on the graph? (g) What is the interval of convergence of your power series? What is the radius of convergence?
Transcribed Image Text:4. In this problem we'll figure out how to represent the function h(x) = ln(1-x) as a power series, (a) Find the derivative of h(r). (Remember there is a little chain rule involved here.) h'(a) = (b) We can recognize h'(x) as similar to the expression we see in the geometric series formula. What is a? What is r? Use this to represent h'(x) as a power series. -1 T=\ -X h'(x) = (c) Find the antiderivative of your power series (term by term) to obtain a power series representation of h(r) itself. h(x) = ln(1-x) = (d) Plug in x = 0 to solve for the value of C. + C (e) Using Desmos or your graphing calculator, graph h(r). Also graph that power series partial sum that includes the first six terms. Are your graphs close to each other? For what interval of r-values is this a good approximation? (f) Expand your power series partial sum to include more terms. What do you notice on the graph? (g) What is the interval of convergence of your power series? What is the radius of convergence?
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