Approximation of an Infinite Series // python

images:

https://media.cheggcdn.com/media%2F2da%2F2dac5ac4-8c92-4d30-ac1f-5a5ae043f9b7%2Fphp0xCrPL.png

https://media.cheggcdn.com/media%2F169%2F169bc149-cc64-468b-bd58-69b992d37348%2FphpFBmqhN.png

https://media.cheggcdn.com/media%2Fa91%2Fa91888a1-b14a-4ac2-a066-158731a52c13%2Fphp4K2J99.png

Recall that a function f(x) can be expressed as a Taylor Series centered at a number zo: o0f(n) f(x) n=0 which can be truncated at any integer N (with varying levels of accuracy). Typically, the closer z is to zo, the better the truncated series will approximate the value of f(x) Below is a plot of the function sin(z), the function f(z)-z, and the n 3 Taylor series expansion of sin) Taylor Series approximations of sin() 5.0 2.5 0.0 2.5 5.0 7.5 -10.0 sin(x) 12.5 -x Taylor Series: n = 3 2 4 Often in science, f(x) -x is used as an approximation for sin(x). You can verify that this is the first nonzero term of the Taylor series of sin(x) about xo - 0. This estimate is valid only in a small neighborhood near x - 0. We want to examine what happens as we get farther from the center of the Taylor series. For a particular value of x = 0.7: Calculate the n -1 and n 3 Taylor series expansions of sin(a) about xo 0 evaluated at . Calculate the n-3 Taylor series expansion of sin(x) about Zo =-evaluated at Calculate the true value of sin(). Use this value to calculate the relative errors of the approximations. Save the results in err_0_1, err_0_3, and err_pi4_3. Remember that the formula for relative error ispro Verue Remember that the formula for relative ytrue Your code snippet should define the following variables: Name err_1 err 0_3 err_pi4_3 Type floating point number floating point number floating point number Description error when x-0-0 and n-1 error when x 0 0 and n-3 error when x_0 pi/4 and n-3 Recall that a function f(x) can be expressed as a Taylor Series centered at a number zo: o0f(n) f(x) n=0 which can be truncated at any integer N (with varying levels of accuracy). Typically, the closer z is to zo, the better the truncated series will approximate the value of f(x) Below is a plot of the function sin(z), the function f(z)-z, and the n 3 Taylor series expansion of sin) Taylor Series approximations of sin() 5.0 2.5 0.0 2.5 5.0 7.5 -10.0 sin(x) 12.5 -x Taylor Series: n = 3 2 4 Often in science, f(x) -x is used as an approximation for sin(x). You can verify that this is the first nonzero term of the Taylor series of sin(x) about xo - 0. This estimate is valid only in a small neighborhood near x - 0. We want to examine what happens as we get farther from the center of the Taylor series. For a particular value of x = 0.7: Calculate the n -1 and n 3 Taylor series expansions of sin(a) about xo 0 evaluated at . Calculate the n-3 Taylor series expansion of sin(x) about Zo =-evaluated at Calculate the true value of sin(). Use this value to calculate the relative errors of the approximations. Save the results in err_0_1, err_0_3, and err_pi4_3. Remember that the formula for relative error ispro Verue Remember that the formula for relative ytrue Your code snippet should define the following variables: Name err_1 err 0_3 err_pi4_3 Type floating point number floating point number floating point number Description error when x-0-0 and n-1 error when x 0 0 and n-3 error when x_0 pi/4 and n-3