# Thread: unif convergence of series

1. ## unif convergence of series

Hello all,I need a bit of help to proove that
$\sum_{n=1}^{\infty} \dfrac{x}{(1+{x})^n}$ uniformally converges , x in [1,2]

, $n \in \mathbb{N}=1,2,3....$

2. Originally Posted by Kiki
Hello all,I need a bit of help to proove that
$\sum_{n=1}^{\infty} \dfrac{x}{(1+{x})^n}$ uniformally converges , x in [1,2]

, $n \in \mathbb{N}=1,2,3....$
Merely note $\forall x\in [1,2]$ that $\frac{x}{(1+x)^n}\leqslant \frac{1}{2^n}$ and apply Weierstrass.

3. Originally Posted by Drexel28
Merely note $\forall x\in [1,2]$ that $\frac{x}{(1+x)^n}\leqslant \frac{1}{2^n}$ and apply Weierstrass.
Sweet! Thank you vm

4. Originally Posted by Kiki
Sweet! Thank you vm
You better prove that. I could have said that your function is less than $\frac{1}{\pi^{\pi^n}}$ and you still would have believed me.

5. Originally Posted by Drexel28
You better prove that. I could have said that your function is less than $\frac{1}{\pi^{\pi^n}}$ and you still would have believed me.
lol man you are right, when I first saw your answer I did mistakenly thought of it as a correct one cause I thought that the numerator was "1" , but instead it was "x" . (I suppose the same thing did happen to you too?)

So I think that its not correct, because for n=1 (and lets say x=2)

As far as mentioning $\frac{1}{\pi^{\pi^n}}$ do we get this from a Taylor Series?

6. I think I ve found it.

$\frac{x}{(1+x)^n}\leqslant \frac{x^n}{(1+x)^n} = (\frac{x}{1+x})^n \leqslant {(\frac{2}{3})^n}$

Is this ok?

7. Originally Posted by Kiki
I think I ve found it.

$\frac{x}{(1+x)^n}\leqslant \frac{x^n}{(1+x)^n} = (\frac{x}{1+x})^n \leqslant {(\frac{2}{3})^n}$

Is this ok?