Convergence and uniformly convergence question

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In summary, the question is asking for the range of positive values of x for which the series \sum_{n=0}^\infty \frac{1}{1+x^n} is (a) convergent and (b) uniformly convergent. The attempt at a solution involves considering two different intervals of x, [0,1] and (1,\infty), and using comparison tests to determine convergence. However, the question of uniform convergence remains a challenge and the idea of proving the series to be a monotonically decreasing function is suggested.
  • #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?
 
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  • #2
your second comparison is not helpful, try instead
1/(1+x)+1/(1+x^2)+...+1/(1+x^n)<1/x+1/x^2+...+1/x^n=(x^(n-1)-1)/(x^n-x^(n-1))

It makes not sense to compare x and n the way you do, because n takes an infinite number of values.

Convergence is uniform if N can be chosen without regard to x.
 
  • #3
Thanks, lurflurf. Your idea was of a great help. But the uniform convergence is still giving me headache. I was thinking that if I could prove that the sum [tex]\sum \frac{1}{1+x^n}[/tex] is a monotonically decreasing function, for any value of n, then the bigger the x, smaller the sum, so It would be necessary less terms for establishing the convergence, and this way, this series wouldn't be uniformly convergent. What do you think?
 

1. What is convergence in mathematics?

Convergence in mathematics refers to the tendency of a sequence of numbers or functions to approach a specific value as the number of terms or iterations increases. It is an important concept in calculus, analysis, and other branches of mathematics.

2. What is the difference between pointwise convergence and uniform convergence?

Pointwise convergence is when a sequence of functions converges to a particular value at each point in the domain. Uniform convergence is when a sequence of functions converges to a particular value at each point in the domain, but also the rate of convergence is the same at all points.

3. How is uniform convergence related to continuity and differentiability?

In general, uniform convergence implies continuity. If a sequence of functions is uniformly convergent, then the limit function will also be continuous. However, uniform convergence does not necessarily imply differentiability. A sequence of differentiable functions can converge uniformly to a non-differentiable limit function.

4. How do you determine if a sequence of functions is uniformly convergent?

To determine if a sequence of functions is uniformly convergent, you can use the Weierstrass M-test. This test compares the sequence of functions to a convergent geometric series and if the terms of the sequence are smaller, the series is uniformly convergent. You can also check for uniform convergence by finding the supremum of the difference between the terms of the sequence and the limit function.

5. What are some applications of uniform convergence in real-world problems?

Uniform convergence has many applications in real-world problems, such as numerical analysis, signal processing, and optimization. For example, in numerical analysis, uniform convergence is used to ensure that numerical methods for solving differential equations will produce accurate results. In signal processing, uniform convergence is used to analyze the convergence of Fourier series and the stability of filters. In optimization, uniform convergence is used to determine the convergence of iterative algorithms for finding optimal solutions.

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