# Homework Help: Convergence of a series.

1. Feb 3, 2006

### 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 im have trouble starting on.. please help.
thank you,
sorry i havnt quite mastered latex

Last edited: Feb 3, 2006
2. Feb 4, 2006

### arildno

(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.

Last edited: Feb 4, 2006
3. Feb 4, 2006

### Pearce_09

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