Valid application of Weierstrass Test?

In summary, the conversation discusses the relationship between the uniform convergence of a series and its absolute uniform convergence. The interlocutors consider different tests, such as the Weierstrass Test and the triangle inequality, and ultimately conclude that if a series is uniformly Cauchy, then the other series is essentially uniformly Cauchy as well.
  • #1
end3r7
171
0
It would seem so

Homework Statement



If
[tex]
\sum\limits_{n = 1}^{\inf } {|{f(n,x)}|}
[/tex] is uniformly convergent on [a,b], then is [tex]
\sum\limits_{n = 1}^{\inf } {{f(n,x)}}
[/tex] uniformly convergent.


Homework Equations





The Attempt at a Solution



I said yes. And just applied the Weierstrass Test with |f(n,x)| <= |f(n,x)| (a basic comparison test)

Should still be valid right? Since the absolutely value is Uniformly Cauchy
 
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  • #2
I don't follow your argument.

Just because a series converge uniformly does not mean it satisfy Weierstrass' M test
 
  • #3
Well, forget the Weierstrass test then...
If one is uniformly Cauchy, wouldn't that make the toher essentially uniformly Cauchy as well?
 
  • #4
This sounds right! (triangle inequality)
 
  • #5
How does uniformly Cauchy apply to absolute uniform convergence
 

Related to Valid application of Weierstrass Test?

What is the Weierstrass Test and how is it used in scientific research?

The Weierstrass Test is a mathematical tool used to determine the convergence of infinite series. In scientific research, it is often applied to analyze the convergence of numerical solutions to mathematical models or to evaluate the accuracy of experimental measurements.

What are the assumptions of the Weierstrass Test?

The Weierstrass Test assumes that the series being evaluated is composed of positive terms and that it is monotonically decreasing. Additionally, it requires the series to be convergent, meaning that the sum of all terms in the series must approach a finite limit as the number of terms increases.

How do you determine if a series converges using the Weierstrass Test?

To determine if a series converges using the Weierstrass Test, you must first find a convergent series that is greater than or equal to the original series. Then, if the larger series converges, the original series must also converge. If the larger series diverges, the original series may still converge, so further analysis is needed.

What is the relationship between the Weierstrass Test and other convergence tests?

The Weierstrass Test is a more general form of the comparison test and the limit comparison test. It can also be used in conjunction with these tests to determine convergence. Additionally, the Weierstrass Test is often used as a preliminary test in the Ratio Test and the Root Test.

What are some potential limitations of the Weierstrass Test?

One potential limitation of the Weierstrass Test is that it may not work for series with alternating signs or for series with alternating terms that do not decrease in magnitude. Additionally, it may not always be easy to find a larger, convergent series to use for comparison. Finally, the Weierstrass Test only determines convergence, not the exact value of the sum, so additional methods may be needed for more precise calculations.

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