Is 1/x Integrable on Intervals Containing 0?

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Homework Help Overview

The discussion revolves around the integrability of the function 1/x on intervals that include the point 0, with a focus on analysis concepts related to improper integrals.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore whether the behavior of the anti-derivative ln(x) at 0 is sufficient to conclude non-integrability. There are questions about the necessity of using partition arguments to demonstrate this property.

Discussion Status

The conversation is ongoing, with participants offering different perspectives on how to approach the proof of non-integrability. Some guidance has been provided regarding the need to consider partitions and upper sums, but no consensus has been reached.

Contextual Notes

There is a specific focus on the requirement to show that 1/x is not improper Riemann integrable, which adds a layer of complexity to the discussion.

robertdeniro
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Homework Statement



Prove that 1/x is NOT integrable on any intervals containing 0

Homework Equations





The Attempt at a Solution



would it be sufficient to say that the anti-derivative, ln x, blows up at 0? would this answer be rigorous enough for an analysis course?

or do i have to use some kind of partition argument?
 
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You mean you can't say that since 1/x AND ln(x) do not exist at 0, it cannot be integrated over 0?
 
I believe you have to show that you can find a partition such that the upper sum is greater than any alpha ( > 0).

Basically, you show that the integral over zero can get as large as you want.
 
sorry, i have to show that its not improper riemann integrable
 

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