# fourier series Integration

After integrating this function to create a fourier series I was confused by why are they raised to the power of k? How does cos(kx) or sin(kx) for example become the number (-1 or 1 in this case) raised to power k? Is this a trigonometry rule?
My second confusion is the last one 4cos(kx)/pi*k^3 specifically the part x was supposed to be replaced by 0 as shown in the square brackets.. why is it just - 1 (the last term!) and not (-1) ^k? I'd appreciate if there was any link reccomendation of a video explaining this because idk what to search for :// Thank you so much in advance )
It just reprents cos(k*pi) so -1,1,-1,...

Second one is similar so you get a cos(k*pi) again from the upper limit, but the lower limit is cos(0) = 1.
(edited 1 year ago)
Original post by mqb2766
It just reprents cos(k*pi) so -1,1,-1,...

Second one is similar so you get a cos(k*pi) again from the upper limit, but the lower limit is cos(0) = 1.

Thank you! Why are the - 1's raised to power k though? Is it just a standard rule that is applied everytime? (except in the case for cos(0) is just 1 and not 1^k?)
Original post by sarah630
Thank you! Why are the - 1's raised to power k though? Is it just a standard rule that is applied everytime? (except in the case for cos(0) is just 1 and not 1^k?)

Its just a representation of an alternating sequence -1,1,-1,1,... You could think of it as a geometric sequence with r=-1, so the signs flip as you march along the sequence. Obviously 1^k = 1,1,1,1,..

You could write 1 = 1^k, but cant see it would help.
(edited 1 year ago)
Original post by mqb2766
Its just a representation of an alternating sequence -1,1,-1,1,... You could think of it as a geometric sequence with r=-1, so the signs flip as you march along the sequence. Obviously 1^k = 1,1,1,1,..

You could write 1 = 1^k, but cant see it would help.

Aahh right. Thank you so much