Hi,
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.
Thanks
i was thinking about the Weierstraβ M-Test, but i cannot think of a nice upper bound.
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
it is somewhat different from the series one, and as you can see, the theorem is talking about a series. it turns out that the series wont uniformly converge if the sequence of functions it is summing does not. the theorem was proven in your book as an example, look at it
First I wanted to say not because it does not converge at . But it does converge (in fact uniformly as shown) on .
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.