Is there any trigonometric function that cannot be written as an infinite series?

Question

Let $p_n(x)=x^n$ for $x\in \Bbb{R}$ and let P=span$\{p_0,p_1,p_2,p_3\dots\}$. Then-

1. P is the vector space of all real valued continuous functions on R

2. P is a subspace of all real valued continuous functions on R.

3. The set $\{p_0,p_1,p_2,p_3\dots\}$ is linearly independent in the vector space of all continuous functions on R
4. Trigonometric functions belong to P

I can easily see option 1. is false and option 2. and 3. are correct but I am not sure about option 4.

Why would any trigonometric function not belong P= span $\{p_0,p_1,p_2,p_3\dots\}$

Answer 1

Some functions cannot be written as Taylor series because they have a singularity for example $f(z) = \frac{1}{1-\cos z}$.

Answer 2

Your $P $ does not contain any infinite series. The span is the set of linear combinations of the $p_n $; in fact whoever wrote the question called it $P $ because it is the set of polynomials.

So the question is whether any trigonometric function is a polynomial. The answer is no, because a polynomial cannot be periodic.