Solution
= Solution
The finite <geometric series> gives $F_n(x)=(1-x^{n+1})/(1-x)$, so the <pointwise limit> is $F(x)=1/(1-x)$. The error is $x^{n+1}/(1-x)$, unbounded as $x\uparrow1$ for every fixed $n$. Thus \b[convergence is not uniform on $(0,1)$]. It is uniform on any smaller interval bounded away from one, since the error is at most $b^{n+1}/(1-b)$ for $0<x\leq b<1$.