Please answer and show work for number 6. Thank you! Recall that the Taylor seri
ID: 2861684 • Letter: P
Question
Please answer and show work for number 6. Thank you!
Recall that the Taylor series sigmma^infinity_k=0 x^k/k! converges to e^x for all real numbers x. In this project we extend this result to complex values of x, with suiprising and beautiful outcomes. The calculations are formal, which means that we work with power series without having proved their convergence. In fact. using advanced techniques, it can be shown that all the series we use in this project converge for all complex numbers. Recall that the imaginary number i is defined by the property i^2 = - 1. Show that i^3 = -1 and i^4 = 1. What is the value of i^12? what is the value of i^15? Assume that x is a real number and replace x by ix in the Taylor series for e^x. Then collect the real and imaginary parts of the series, identify the Taylor series for cos x and sin x, and derive Euler's formula: e^ix = cos x +i sin x. Use Euler's formula to evaluate the following expressions: e^ipi e^i2pi e^ipi/2 e^ipi/4. The first of these expressions, e^ix + 1 = 0. is the remarkable Euler's identity. which contains five of the fundamental constants of mathematics (1,0, i, e, pi). Show that e^-ix = cos x - i sin x.Use Euler's formula and step 4 to show that cos x = e^ix + e^-ix/2 and sin x = e^ix - e^-ix/2 Let x and y be real numbers and confirm that e^x + iy = e^x (cos y +1 sin y).Explanation / Answer
use result from ans 2. e^iy=(cosy+isiny)
e^(x+iy)=(e^x)(e^iy)=e^x(cosy+isiny)