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?

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...

See similar textbooks

Related questions

Q: o I find the growth/d

A: Here we have , for the following exponential function :- x 0 1 2 3 f(x) 5 10 15 20

Q: Example 1: A cattle farm with a population of 240 cows is declining by 3.4% each year. Determine a…

Q: The population (P) of a certain town is said to double over a twenty-five year period and its…

A: To determine the yearly growth of the population

Q: Find the derivative of the following function. [In(2*)]* O 24|In(2*)]* 24|In (2*)]* 24|In(z")]* 24|…

Q: 1.) Simplify the following: (a) (ln 2 + ln 3)/ ln 36 (b) e^−2 ln x (c) ln(e(^−x^2)) 2. Let f(x) =…

A: 1. (a) ln2+ln3ln36=ln2×3ln62=ln62×ln6=12=0.5 b) e-2lnx=elnx-2=x-2=1x2 c) ln(e-x2)=-x2 SInce,…

Q: In each case below, select the best answer to the question. (a) The graph of f(x) is shown below.…

A: Note:- Since you asked multilple subparts question,according to our policy we can provide first…

Q: If y' = x√(ey) + (y/x) Then, e(y/x) = ? a) (-1 / Cln(x)) b) (-1 / C + ln(x)) c) (C / 1 + ln(x))…

Q: per day? 57. Drugs in the bloodstream The concentration y of a certain drug in the bloodstream t…

Q: • Find f(22) (0), for f(0) = 2 cos(30), using pattern recognition. (Don't take 22 derivatives!) What…

A: The given function is fθ=2cos3θ.

Q: Find h'(-3) if h(x)=4g(x)-5f(x), f(-3)-7,and g'(-3)=-2. Oh'(-3)-0

Q: Set up the formula to find the balance after 8 years for a total of $12,000 invested at an annual…

Question

100%

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?

AI-Generated Solution

AI-generated content may present inaccurate or offensive content that does not represent bartleby’s views.

Unlock instant AI solutions

Tap the button
to generate a solution

Click the button to generate
a solution

SEE SOLUTION

Similar questions

- please solve this on a piece of paper so it will be easier to understand.
help me with question 3
Question 3 find f'(x) and simplify. f(x) = x(In x)² A. (2+In(x))-In(x) B. 1+2ln(x) c. 2ln(x) XI D.2ln(x)
Suppose a company's earnings are given by E(x) = P(x) + I(x), where x is the number of years since 2000, P(x) is the total profit from 2000 to year x, and I(x) is the intangible growth (the growth in value of the company's intangible assets such as its good name). If P(x) = 1.5x + 1 and 1(x) = 0.15x + 3 for a certain company, determine the average earnings formula (earnings per year since 2000). Answer How to enter your answer (opens in new window) 2 Points A(x) = Keypad Keyboard Shortcuts
(a) If h(x) = f(x)g(x), we know h' = f'g+fg'. Take another derivative and show that h" = f"g+2f'g' + fg". (Show means, do the computation and then notice that h" is indeed what it is supposed to be.) Show your work. (b) Find a similar formula for h"". Show your work.
2. A population of deer is increasing at a rate of 5% a year. At t = 0 years, there are 98 deer in the population. (a) Using the information above, write a formula for the deer's population P = f(t). (b) Using your formula from (a), determine the approximate number of deer after 5 years have passed.
Which of the following sequences converge? Give reasons for your answers.