Fourier series for t

[patter2809]

Q. Find best approximation to f(t) = t in interval (-pi,pi) by a linear combination of exp(kit), for integer k, and represent as a real-valued function. Write function as a sum of functions of sin(kt), and cos(kt) for k > 0.

A. Calculated Fourier coefficients to be i/k when k is even, -i/k when k is odd, and pi/2 for k = 0.

P_n(t) = pi/2 + Σ_k=-nⁿ (-1)^k i e^ikt / k. But using e^ikt+e^-ikt = 2cos(kt), the real part of P_n(t) is just pi/2 as Σ_k=-nⁿ (-1)^k i e^ikt/k = Σ_k=1ⁿ (-1)^k2icos(kt)/k

Please help, this is driving me mad!

 

[JohnZ]

a0 = 1/pi Integral (from -pi to pi) t dt
ak = 1/pi Integral (from -pi to pi) t * cos(kt) dt
bk = 1/pi Integral (from -pi to pi) t * sin(kt) dt

fourier series:
a0/2 + summation (from k=1 to infinity) (ak * cos(kx) + bk*sin(kx))