Fourier Series and geometric sum at the end

[EnigmaTuring]

I'm having difficulty in converting the last step of my Fourier Series into a geometric series.

On one question I got to the step:

(1/i(n-k)).e^(i(n-k)Theta) integrated between Pi and -Pi but why is this equal to ((-1)^(n-k)-(-1)^(n-k))/i(n-k)?

...............................

On my second second question I got 2/Pi.((Cos n Theta)/ n^2) Integrated between Pi and zero but again why is this equal to 2/Pi.((-1)^(n)-1)/n^2)

Can someone explain how I calculate these please? Hope it's clear enough.

 

[Ishuda]

I'm having difficulty in converting the last step of my Fourier Series into a geometric series.

On one question I got to the step:

(1/i(n-k)).e^(i(n-k)Theta) integrated between Pi and -Pi but why is this equal to ((-1)^(n-k)-(-1)^(n-k))/i(n-k)?

...............................

On my second second question I got 2/Pi.((Cos n Theta)/ n^2) Integrated between Pi and zero but again why is this equal to 2/Pi.((-1)^(n)-1)/n^2)

Can someone explain how I calculate these please? Hope it's clear enough.

Are you sure that your functions are not the result of an integration and should be evaluated there. For example
\(\displaystyle cos(n\, \theta)|_0^{\pi} = cos(n\, \pi)\, -\, cos(0)\, =\, (-1)^n\, -\, 1\)