- #1
thesaruman
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Homework Statement
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
Homework Equations
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?