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# Taylor Series, Matrix Algebra and Complex Numbers

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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.