to show this function does not belong to the uniform closure of A Problem no: $21$ 
Chapter $7$: Sequence and series if functions
Book: Principles of mathematical analysis 
Writer : W. Rudin
Let $K$ be the unit circle in the complex plane and $A$ be the algebra of all functions of the form 
 $f( e^ {ia}) = \sum_{n=0}^{n=N} c_{n} e^{i n a}$ 
Then A seperates points on $K$ and $A$ vanishes at no point of $K$ , but there are continuous functions on $K$ which are not in the uniform closure of $A$.
I have been able to do all the things except the last part.
I am unable to find a continuous function on $K$ which does not lie in the uniform closure of $A$.
I saw the answer is $f(z)= \bar z$ which is clearly continuous.
My question is how to show this function does not belong to the uniform closure of $A$.
Any help would be appreciated.
Thanks in advance.

Prove that if $f$ is an element of the uniform closure $A$ then $\int_0^{2\pi}f(e^{it})e^{it}\,dt=0$.
