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Finding an integral using a series

  • #1
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Homework Statement


##\displaystyle \int_0^1 \frac{\arctan x}{x}dx##

Homework Equations




The Attempt at a Solution


I converted the integral to the following; ##\displaystyle \int_0^1 \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+1}dx##. In this case am I allowed to swap the summation and integral signs?
 

Answers and Replies

  • #2
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5,004

Homework Statement


##\displaystyle \int_0^1 \frac{\arctan x}{x}dx##

Homework Equations




The Attempt at a Solution


I converted the integral to the following; ##\displaystyle \int_0^1 \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+1}dx##. In this case am I allowed to swap the summation and integral signs?
Since the series is absolutely convergent on the interval in question, yes, you can exchange the operations. Here's a link to another thread that discusses this -- https://www.physicsforums.com/threads/interchanging-summation-with-integral-differentiation-with-integral.564620/.
 
  • #3
Ray Vickson
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Homework Helper
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Since the series is absolutely convergent on the interval in question, yes, you can exchange the operations. Here's a link to another thread that discusses this -- https://www.physicsforums.com/threads/interchanging-summation-with-integral-differentiation-with-integral.564620/.
This argument does not quite work: the series is absolutely convergent on the open interval ##0 \leq x < 1## but is only conditionally convergent at ##x=1##. However, the OP can try another argument instead: he can check for uniform convergence on the closed interval, and that will be enough. See, eg.,
https://www.dpmms.cam.ac.uk/~agk22/uniform.pdf .
 
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  • #4
33,310
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This argument does not quite work: the series is absolutely convergent on the open interval ##0 \leq x < 1## but is only conditionally convergent at ##x=1##. However, the OP can try another argument instead: he can check for uniform convergence on the closed interval, and that will be enough. See, eg.,
https://www.dpmms.cam.ac.uk/~agk22/uniform.pdf .
I was a bit worried about that endpoint at x = 1, but didn't include my concern in what I wrote.
 

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