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To start off, It's worth noting the definition of eigenvalues: for a Matrix A (n x n), it's ith eigenvalue (λ_i) is defined as the scalar constant the ith eigenvector (v_i) is multipli...

The general solution y_GS(t) to a differential equation in the form ay''(t)+by'(t)+cy(t)=f(t) is the sum of the complementary function y_CF(t) and a <...

Need whiteboard throughout to properly answer, so will go through the ideas of what to do:

Take the base case of when n=1, and show that            (sum from 1 to 1) j = n(n+1)/2       is equal to ...

Simplify i⁴?

We know that i = -1^1/2

Therefore i²=-1

As i⁴=i²xi² 

Then i⁴=-1x-1=1

C + iS = (acos(x) + a^2cos(2x) + a^3cos(3x) + ...) + i( asin(x) + a^2sin(2x) + a^3sin(3x) + ...)

= a(cos(x) + isin(x)) + a^2(cos(2x) + isin(2x)) + a^3(cos(3x) + ...