Weierstrass M-Test: Show Uniform Convergence on -infinity<x<infinity

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In summary, you can show that Sigma from 3 to infinity of 1/(n^2+x^2) is uniformly convergent on -infinity< x<infinity by finding a convergent series, such as 1/n^2, that is greater than or equal to the given function and using the M-test.
  • #1
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How do I show that Sigma from 3 to infinity of 1/(n^2+x^2) is uniformly convergent on -infinity< x<infinity using the M-test? Can anyone help? Thanks in advance.
 
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  • #2
math&science said:
How do I show that Sigma from 3 to infinity of 1/(n^2+x^2) is uniformly convergent on -infinity< x<infinity using the M-test? Can anyone help? Thanks in advance.

Well, you need to find terms [itex]M_n[/itex] with [itex]|1/(n^2+x^2)|\leq M_n[/itex] for all x, such that:
[tex]\sum_{n=3}^{\infty}M_n[/tex] is convergent.

Looking at your function, does any series come to mind?
 
  • #3
1/n^2? That's what I thought of initially. Is that right and that simple?
 
  • #4
Why the doubt?
Is [itex]1/(n^2+x^2)
\leq 1/n^2[/itex]?
Is [itex]\sum_{n=3}^{\infty} 1/n^2[/itex] convergent? If so, then according to the M-test your series is uniformly convergent. It's that simple.
 

1. What is the Weierstrass M-Test?

The Weierstrass M-Test is a method used to determine whether a series of functions converges uniformly on a given interval. It was developed by German mathematician Karl Weierstrass in the 19th century.

2. How does the Weierstrass M-Test work?

The Weierstrass M-Test states that if a series of functions, fn(x), satisfies two conditions:

  • There exists a sequence of real numbers, Mn, such that for all x in the interval, |fn(x)| ≤ Mn for all n
  • The series ∑Mn converges
then the series of functions, fn(x), will converge uniformly on the interval.

3. What is the significance of uniform convergence?

Uniform convergence means that as the number of terms in the series increases, the functions get closer and closer to their limiting function at the same rate at every point in the interval. This is important because it ensures that the series of functions will have the same limiting function, regardless of the order in which the terms are added.

4. How is the Weierstrass M-Test used to show uniform convergence on -infinity<x<infinity?

To show uniform convergence on -infinityn, and prove that the series ∑Mn converges. Then, we must show that for any x in the interval, |fn(x)| ≤ Mn for all n. If these conditions are met, we can conclude that the series of functions converges uniformly on -infinity

5. What are some examples of using the Weierstrass M-Test to show uniform convergence on -infinity

The Weierstrass M-Test can be used to show uniform convergence on -infinityx) converges uniformly on -infinityn = 1/n2, since the series ∑(1/n2) converges. Therefore, we can conclude that the series of functions, fn(x) = 1/nx, converges uniformly on -infinity

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