Show that the series converges uniformly on [0,a] for any a in (0,1)
There is a theorem in the book that says that is the series of functions satisfies the cauchy criterion uniformly on a set S, then the series converges uniformly on S. So I guess i'm supposed to use that. I'm not sure how to do a cauchy criterion proof though.
use the theorem that says: If the series converges uniformly on a set S, then
and use the contrapositive
TPH should be able to come up with a more elegant solution though, i'm sure he can use the Cauchy criterion proof/disproof
1)The functions are uniformly continous functions on .
2)Therefore the series of functions is uniformly continous on .*
3)Thus, if is Cauchy in then is also Cauchy (properties of uniformly continous functions). But as you can see as you get closer to the value of becomes unbounded. So it cannot be Cauchy.
4)Thus, our initial assumption that the convergence was uniform on the full interval was wrong.
*)Because uniformly continous functions that converge uniformly will convergence to a uniformly continous functions.