# Taylor Series, Matrix Algebra and Complex Numbers

A. Basic Mathematics - Please view the attachment for proper formatting of this question.

1. Given that the Taylor series for the function

1/(1-x) = 1 + x + x^2 + x^3 + ...,

use this to show the following

x/(1+x^2)^2 = x - 2x^3 + 3x^5 - ...

log(1-x) = -x - x^2/2 - x^3/3 - x^4/4 - ...

2. Consider complex number division (a + ib)/(c + id) which we express as re^ia. Using Euler's identity only, work out the precise form for the quotient's modulus r and argument alpha. Your calculations should not involve division of complex numbers.

3. Using row operations evaluate the following determinant

y-z z-x x-y

z-x x-y y-z

x-y y-z z-x

4. Find the eigenvalues and eigenvectors of the following matrix

3 3 3

3 -1 1

3 1 -1

Verify that the eigenvectors are mutually orthogonal and hence diagonalise the matrix. Show all working.

Â© BrainMass Inc. brainmass.com March 4, 2021, 8:45 pm ad1c9bdddfhttps://brainmass.com/math/linear-algebra/taylor-series-matrix-algebra-and-complex-numbers-192119

#### Solution Summary

This solution looks at taylor series, matrix algebra and complex numbers in an attached Word document.