- #1

Oxymoron

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My defintion of an absolutely convergent series is one in which you can rearrange the series and it converges to the same value, i.e.

[tex]\sum |x_n| < \infty[/tex]

My question is: If one has a double series [itex]\sum x_{m,\,n}[/itex] which is absolutely convergent [itex]\sum |x_{m,\,n}| < \infty[/itex] then can I apply Fubini's Theorem to conclude that

[tex]\sum_n(\sum_m x_{m,\,n}) = \sum_m(\sum_n x_{m,\,n})[/tex]

I ask this because when we covered Fubini's Theorem in class we spoke of being able to change the order of integration. So I thought, since integration is similar to sums, why can't we change the order of a sum using Fubini's Theorem?

[tex]\sum |x_n| < \infty[/tex]

My question is: If one has a double series [itex]\sum x_{m,\,n}[/itex] which is absolutely convergent [itex]\sum |x_{m,\,n}| < \infty[/itex] then can I apply Fubini's Theorem to conclude that

[tex]\sum_n(\sum_m x_{m,\,n}) = \sum_m(\sum_n x_{m,\,n})[/tex]

I ask this because when we covered Fubini's Theorem in class we spoke of being able to change the order of integration. So I thought, since integration is similar to sums, why can't we change the order of a sum using Fubini's Theorem?

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