Does a uniformly convergent sequence imply a convergent series

[gottfried]

Does anybody know if this statement is true?

$\sum$ f_n converges absolutely and uniformly on S if ( f_n) converges uniformly.

Also if R is the radius of convergence and |x|< R does this imply uniform and absolute convergence or just absolute convergence.

 

[HallsofIvy]

Suppose f_n converges uniformly to something other than 0.

The phrase "radius of convergence" only applies to power series and, in that case, the series converges both absolutely and uniformly inside the radius of convergence.