(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For what range of positive values of x is [tex]\sum_{n=0}^\infty \frac{1}{1+x^n}[/tex]

(a) convergent

(b) uniformly convergent

2. Relevant equations

3. The attempt at a solution

I didn't figure out how to separate convergence and uniformly convergence for this series.

My idea was to consider two different intervals: x in [0,1] and in (1,[tex]\infty[/tex]).

For the first interval,

[tex]\frac{1}{1+x}+\frac{1}{1+x^2}+\cdots + \frac{1}{1+x^n}\ge \frac{n}{1+1}[/tex];

and therefore, is divergent.

For the other one, I considered a similar idea,

[tex]\frac{1}{1+x}+\frac{1}{1+x^2}+\cdots + \frac{1}{1+x^n}\le \frac{n}{1+x}[/tex];

but for this series be convergent, is necessary that x > n.

So, please, can anyone give a little help here?

How do I decide about the convergence? And uniform convergence?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Convergence and uniformly convergence question

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