If f(x) := {n=0 (-1)"x" 3” n2 for x € (-3, 3), find f(100) (0). Hint:...
5. A function f has Taylor series (at 0) f(x)=0+2x+ 4x2/2! + 3x3/3!+... Assume f−1 exists. Find as much of the Taylor series of f−1 (at 0) as you can. (Since you only know the first few terms of the Taylor series for f, you can only figure out f−1. (Hint: There are two ways of doing this problem. One is get the derivatives of f−1 from knowing the derivatives of f; we talked about the first derivative in January...
Can someone walk me through how to do question 2 with all the
proper work shown?
Horne, vork # 3 MİATH 1206 Show all work! 1. (10 pts) Find the Taylor series expansions for f(x) = sin at z = 0 and x = 3, Find the radius of convergence for these series. 2. (5 pts) Find the Taylor series expansion for f(x) = 1/z at 2. 3. (5 pts) Find the sum of the serics rA 5nn! 4" (5...
2. The Taylor series of the function f(x) = - iſ about x = 0 is given by (x − 2)(x2 – 1) 3 15 15 2. 63 4 F=3+ = x + x2 + x + x4 + ... (x − 2)(x2 - 1) 8 16 6 (a) (6 marks) Use the above Taylor series for f(x) = . T and Calcu- (x − 2)(x2 – 1) lus to find the Taylor series about x = 0 for g(x)...
(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.
The function f(x) = - - may be represented by the power series 1-x Part 1: Compute Some Coefficients Find the first four coefficients for the power series: MMMM Part 2: What's the Pattern? Part 3: Radius of Convergence
Let f(z-y2 sin(x+-) Answer the following. Show and explain your work. (a) Find the Taylor polynomial of order 4. generated by f(x) at zo (b) Describe the MacLaurin series of f (with or without the sigma notation). (Hint: What pattern do the derivatives of f at z-0 follow?) (c) Does the MacLaurin series of f converges absolutely, converges conditionally or diverges at -1?
Let f(z-y2 sin(x+-) Answer the following. Show and explain your work. (a) Find the Taylor polynomial of...
3 is represented as a power series: (1 point) The function f(x) 1+36x2 Σ f(x) - n-0 Find the first few coefficients in the power series. CO CI C2 C3 CA Find the radius of convergence R of the series R =
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch |Dml, the magnitude of the Fourier series coefficients vs.o in the range of -4s n S4 c.) Evaluate and sketch the phase angle of D, vs. co in the same range (-4S n S4) d.) Find the signal average power e) Find the approximate average power of...
3. (25 Points) Find f(t). f(0) + f(t - 1)f(t)dt = t. Hint: The second term on the left side is a convolution and it might be helpful to use the Laplace Transform. 1 4. (10 Points) Solve the initial value problem by Laplace transform techniques. x" + 5x' + 4x = 0;x(0) = 1,x'(0) = 0. I 5. (15 Points) Find a series solution for the following differential equation. Calculate the radius of convergence. 2(x - 1)y' = 3y...
Solve the Taylor Series.
1. (a) Use the root test to find the interval of convergence of-1)* に0 (b) Demonstrate that the above is the taylor series of f()- by writing a formula for f via taylor's theorem at α-0. That is write f(x)-P(z) + R(x) where P(r) is the nth order taylor polynomial centered at a point a and the remainder term R(x) = ((r - a)n+1 for some c between z and a where here a 0. Show...