Discrete Fourier series of an odd signal

Question

Assuming the signal shown below :

I have found an expression for fourier series coeffecients as the following: $$a_{k} = \frac{1}{5}+\frac{j}{5}\sin{\frac{2\pi}{5}k}$$

Which matches with what the books suggests as an answer. My confusion is this : why are they not purely imaginary ?

I tried to prove this property the following way: Signal is real and even: $$\implies x[n] = x^{*}[n] = -x[-n] $$
with * denoting the conjugate. And using properties of conjugation, time-reversal and linearity we get : $$a_{k} = a^{*}_{-k} = -a_{-k} $$

Then I concluded : $$a_{k} = -a_{-k} \implies a_{k}\ are\ odd\ in\ k$$ $$ a^{*}_{-k} = -a_{-k} \implies a_{k}\ are\ purely\ imaginary $$

What is wrong with all the above? Is any step in my reasoning wrong or is the solution to the problem incorrect ?

Answer 1

You are correct that a purely odd signal should have purely imaginary coefficients. Your signal, however, is not odd -- it is the sum of an odd part (most of the samples), and an even part (the nonzero sample at time = 0).