Is there any nonzero infinite sum of polynomials that converges to zero? Is there any infinite series of polynomials of the form 
$\sum_{n=0}^\infty a_n x^n$ that converges to zero on the interval $[0,1]$ where at least some $a_i \neq 0$? 

You do not say anything about the meaning of convergence (pointwise, uniform, $\dots$ ?).
Be that as it may, the answer is "yes". Let $p_n(x) = (1/n) x$. Then $p_n(x) \to 0$ uniformly.
