Fourier series - complex exponential form

Hi,

I was taking a break from numerical analysis and had a sneak peak at Fourier Series.

I want to rewrite the trigonometric form into the form with complex exponentials.

I have,

$\displaystyle f(t) = \sum^n_{k=0}a_ksin(2\pi kt) + b_kcos(2\pi kt)$,

and rewrite it using,

$\displaystyle sin(2\pi kt) = \frac{e^{2\pi ikt}-e^{-2\pi ikt}}{2i}$,

and,

$\displaystyle cos(2\pi kt) = \frac{e^{2\pi ikt}+e^{-2\pi ikt}}{2}$ to get,

$\displaystyle f(t) = \sum^n_{k=0} \frac{e^{2\pi ikt}(a_k+ib_k)-e^{-2\pi ikt}(a_k-ib_k)}{2i}$.

Not sure where to go from here. I see that there is come complex conjugate action in the numerator...

Any hints?

Thanks