In Reply to: Question about HW5 Q2e posted by Josh on May 08, 2003 at 16:39:24:
>Hello,
>Q2e asks to find a power formal power series centered at 0 with radius 1 such that it converges pointwise on (-1,1) but not uniformly on (-1,1).
>According to Theorem 12c - uniform convergence on compacta - For any 0 < r < R the series converges uniformly to some function f on the compact interval [a-r,a+r].
>Using this, then for any epsilon>0 , the series converges uniformly on the interval [-1+epsilon,1-epsilon] so the series is uniformly convergent over any closed interval in (-1,1), therefore there does not exist any such series?
Being uniformly convergent on every interval [-1+epsilon,1-epsilon]
does not imply being uniformly convergent on (-1,1), even though
it is true that if one takes the union of all the intervals
[-1+epsilon,1-epsilon] one gets (-1,1). It is similar to how
a function can be bounded on every interval [-1+epsilon,1-epsilon]
but still be unbounded on (-1,1); consider for instance
the function 1/(1-x).
Basically, uniform convergence is not a "pointwise" property;
knowing it is true on a number of sets does not imply that it
is true collectively on the union of those sets.
Terry