Uniform Convergence of Series on [0,1]

[Pearce_09]

I wish I had more done.. but I'm have trouble understanding this...

Show that the series The sum is k=0 to infinity $$\sum^\infty_0 1-x/(2+x)^k$$ converges uniformly on [0,1].

What I have done so far is take the
the lim is k-->infinity $$\lim_\infty 1-x/(2+x)^k = 0$$
therefore the limit exists...
does this even matter.. i mean, do I even need to state this.. because I still need to say if it converges on [0,1] which I am have trouble starting on.. please help.
thank you,
sorry i havnt quite mastered latex
adam

 

[arildno]

Sloppiness with your parentheses! :grumpy:
(I take it you meant $(1-x)/(2+x)^{k}=\frac{1-x}{(2+x)^{k}}$)

Hmm.. what you have shown so for is that for any fixed x, the series does not necessarily diverge.

As for proving uniform convergence, split it in two:

1. Show that you have pointwise convergence.
2. Having 1, since you have pointwise convergence on a compact set, then you also have uniform convergence on that set.

 

[Pearce_09]

this is just a test
suppose $$|f_k(x)| \leq M_k$$
then, $$\sum^\infty_1 |f_k(x)| \leq \sum^\infty_1 M_k$$