Absolutely convergent but not uniformly convergent

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In summary, the speaker was able to show that the series converges absolutely for nonzero x by rewriting it in the form \sum_{k=1}^{\infty}x\left(\frac{2}{3}\right)^{k}\frac{sinx}{x}. However, they are unsure about the uniform convergence of the series and are trying to show that for any given epsilon, the tail of the series cannot be bounded. They are attempting to use the Cauchy criterion, but are struggling to prove it.
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I'm trying to show [itex]\sum_{k=1}^{\infty}2^{k}sin(\frac{1}{3^{k}x})[/itex] does not converge uniformly on any (epsilon, infinity)

now I was able to show that it converges absolutely for x nonzero, by getting it in the form [itex]\sum_{k=1}^{\infty}x\left(\frac{2}{3}\right)^{k}\frac{sinx}{x}[/itex] and so the sinx/x < 1 drops out and so its a convergent geometric series. This x is not really x, I just mean its of the form sinx/x. so the series converges to 3x. However I'm not sure about the uniform part, I'm trying to show for any episilon, I can't bound the tail of the series but I don't know that it converges to 3x do I, all I know is that it converges to something smaller than 3x

so I need to bound the tail using cauchy criterion but can't figure out why it's true. ie for n sufficiently large, [itex]\sum_{k=n}^{m}2^{k}sin(\frac{1}{3^{k}x})[/itex] needs to be less than any given [itex]\varepsilon[/itex]
 
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can't get tex to work

You must use XML-like "tags" before and after the LaTex. The tags are "itex" and "/itex" enclosed inside square brackets for inline LaTex and "tex" and "/tex" for LaTex on its own line.

I'm trying to show [itex] \sum_{k=1}^{\infty}2^{k}sin(\frac{1}{3^{k}x}) [/itex] does not converge uniformly on any [itex] (\epsilon, \infty)[/itex]
 

1. What is the definition of absolute convergence?

Absolute convergence is a type of convergence in which the sum of an infinite series converges, regardless of the order in which the terms are added. In other words, the series converges even if the terms are rearranged.

2. How is absolute convergence different from uniform convergence?

Uniform convergence is a type of convergence in which the limit of the function approaches the limit of the sequence of functions uniformly, meaning that the difference between the function and its limit becomes arbitrarily small for all points in the domain. Absolute convergence, on the other hand, does not require the function to approach its limit uniformly.

3. Can a series be absolutely convergent but not uniformly convergent?

Yes, it is possible for a series to be absolutely convergent but not uniformly convergent. This means that the series converges, but the convergence is not uniform, and the limit function may not accurately represent the series.

4. How can you determine if a series is absolutely convergent but not uniformly convergent?

To determine if a series is absolutely convergent but not uniformly convergent, you can use the Cauchy convergence test. If the series passes this test, it is absolutely convergent. However, to determine if it is also uniformly convergent, you will need to use additional techniques such as the Weierstrass M-test.

5. Why is it important to distinguish between absolute convergence and uniform convergence?

It is important to distinguish between absolute convergence and uniform convergence because they have different implications for the behavior and accuracy of the series. Absolute convergence guarantees that the series converges, but it may not accurately represent the limit function. Uniform convergence, on the other hand, ensures that the series accurately represents the limit function, but it may not converge at all points in the domain.

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