Uniform Convergence Series If $\sum_{k=0}^\infty {a_k}x^k$ has a interval of convergence $(-1, 1)$, and $f(x)=\sum_{k=0}^\infty {a_k}x^k $, for $x\in[0,1)$, then the series does not converge uniformly to $f(x)$ on $[0,1)$.
I'm trying to either come up with a proof for this statement.

Hint: We are given $\sum_{k=0}^{\infty}a_kx^k$ diverges at $x=1.$ I.e., $\sum_{k=0}^{\infty}a_k$ diverges. This implies the partial sums of $\sum_{k=0}^{\infty}a_k$ do not form a Cauchy sequence. Perhaps this implies the partial sums of $\sum_{k=0}^{\infty}a_kx^k$ are not uniformly Cauchy on $[0,1)?$
