Can a series of polynomials converge non-uniformly?

Question

Is there an example of a series of polynomials, say, the degree equals the index and converges non-uniformly? In other words, does point-wise convergence of a polynomial series imply uniform convergence?

Answer 1

No, uniform convergence is stronger even for polynomials. In fact, one of the classic examples of a sequence of continuous functions which converges pointwise but not uniformly is a sequence of polynomials: it is $f_n(x)=x^n$ on $[0,1]$. This converges pointwise to a function which is $0$ except at $1$, where it is $1$.

In light of the Weierstrass approximation theorem, if this were not the case, then pointwise convergence of any sequence of continuous functions would imply uniform convergence.