Improper Integral Convergence for f(x)=1/(x^p)

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Homework Help Overview

The discussion revolves around the convergence of the improper integral of the function f(x)=1/(x^p) over the interval from 0 to 1, particularly focusing on the conditions under which this integral converges based on the parameter p. The original poster expresses uncertainty about how to approach the problem, which involves understanding improper integrals and their convergence criteria.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants suggest calculating the integral from a limit approaching 0 and discuss the implications of different values of p on convergence. There are questions about the original poster's understanding of convergence and divergence, particularly regarding the integral's behavior as p varies.

Discussion Status

The discussion is ongoing, with participants providing hints and guidance on how to approach the problem. There is an emphasis on clarifying the conditions for convergence and the nature of the integral as improper. Multiple interpretations of the problem are being explored, particularly regarding the values of p that affect convergence.

Contextual Notes

Participants note that the integral is improper due to potential discontinuities at x=0, which necessitates a limit approach. There is a specific focus on the need to show work and reasoning behind the conclusions drawn regarding convergence.

zachem62
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Homework Statement



Consider the function f(x)=1/(x^p).

When p>1, the integral of 1/(x^p) from 1 to infinity converges.


i) For what values of p does the integral of 1/(x^p) from 0 to 1 converge? (0<p<infinity, p does not equal 1).

ii) Confirm the answer by re-writing the integral of 1/(x^p) from 0 to 1 in terms of area and an integral in terms of y. Comment on any symmetry/asymmetry that this relation demonstrates).

I did a question similar to this but it was much simpler. I have no idea how to even start with this question. PLEASE HELP!


The Attempt at a Solution

 
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Just calculate ##\int_\epsilon^1 x^{-p}\, dx##, let ##\epsilon \rightarrow 0^+## and see which ##p## values work.
 
zachem62 said:
i)(0<p<infinity, p does not equal 1).

Is this your answer to part one? If so, consider that these values will make the integral diverge, not converge. Also, please show us your work.

\int_{0}^{1} x^{-p} dx = ?
 
iknowless said:
Is this your answer to part one? If so, consider that these values will make the integral diverge, not converge. Also, please show us your work.

\int_{0}^{1} x^{-p} dx = ?

No that is not the answer to part one. Everything I have posted is part of the question and I have no clue how to get started and finish the question.
 
zachem62 said:
No that is not the answer to part one. Everything I have posted is part of the question and I have no clue how to get started and finish the question.

Did you read post #2? That gives a clue.
 
On the interval [0,1], which is where we are evaluating the integral, are there any points of discontinuity for values of x?

If so, then the integral is improper, and you must replace that x-value with a variable, then take the limit as that variable approaches the x-value from either the right or the left. Your book will have a specific theorem or example of an improper integral.
 

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