f) a^x 4. Expand the following functions in a Maclaurin series: (2) 1+x (b) (1 +...
5. Expand the following functions as Fourier-Legendre series: (i) f(x)=x3 x >0 x < 0 1, (ii) f(x) = lo, 5. Expand the following functions as Fourier-Legendre series: (i) f(x)=x3 x >0 x
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. (Assume that f has a power series expansion f(x) = cos x Find the Taylor series for f centered at 4 if f(n) (4) = (-1)" n! 3" (n + 1) What is the radius of convergence of the Taylor series?
(d) f(x) = (1 + x) ln(1 + x) Hint: differentiate. (4) Expand the following functions into power series centered at 0 and find the radius of convergence. You can either use geometric series method, known expansions, or derivatives. You don't have to analyze the remainder.
5,9,13,17 1-X 1. What is the difference between a Taylor series and Maclaurin series? 2. T/F: In general, pn() approximates f(x) better and better as n gets larger. 3. For some function f(x), the Maclaurin polynomial of degree 4 is pa(x) = 6 + 3x - 4x + 5x – 7x*. What is p2(x)? 4. For some function f(x), the Maclaurin polynomial of degree 4 is p(x) = 6 + 3x - 4x + 5x – 7x*. What is f"O)?...
1,2,3, and 4 Here are some practice exercises for you. 1. Given f(x) e2, find the a. Maclaurin polynomial of degree 5 b. Taylor polynomial of degree 4 centered at 1 c. the Maclaurin series of f and the interval of convergence d. the Taylor series generated by f at x1 2. Find the Taylor series of g(x) at x1. 3. Given x -t2, y t 1, -2 t1, a. sketch the curve. Indicate where t 0 and the orientation...
Using the Maclaurin series for f(x) = sin x, derive the Maclaurin series for g(x) = x sin 2x 1. Hint: It is not necessary to do any differentiation to do this problem. Using the Maclaurin series for f(x) = sin x, derive the Maclaurin series for g(x) = x sin 2x 1. Hint: It is not necessary to do any differentiation to do this problem.
4. Derive the Maclaurin series for In (1 + x) 5. Derive the Maclaurin series for In (1 – x) 6. Derive the Maclaurin series for In (*) 7. Find the Maclaurin series for x sin x (1-x)
Question 4: Talyor. Maclaurin and Power Series For parts a, b, c and d, use the following function: f(x) = (-3x a) (3 points) Write the Taylor polynomial of degree four for f(x) centered at 0. b) (2 points) Use the Taylor polynomial from part a to estimate the value of e-0.3. (Hint: let find x). c) (3 points) Write the series generated by f(x) at zero in sigma notation. d) (3 points) Find the radius of convergence and state...
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = sin(πx/2)fx = _______ Find the associated radius of convergence R.
19. . 20 . 21 Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = e-3x f(x) = Σ n = 0 Find the associated radius of convergence R. R = Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) = 0.] f(x)...