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How would you Integrate a Sigma Sum?

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- Thread starter Superposed_Cat
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- #1

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How would you Integrate a Sigma Sum?

- #2

arildno

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- #3

Mark44

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I.e., term by term.

- #4

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thanks.

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arildno

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Not necessarily, but usually beneficially!I.e., term by term.

- #6

FeDeX_LaTeX

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what?

- #8

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Exactly. If it's an infinite sum and it doesn't converge uniformly, then we cannot do so term by term. The usual example is from Kresyzig:

Let: [itex]u_m(x)=mxe^{-mx^2}[/itex]

and consider:

[tex]\sum_{n=1}^{\infty} f_n(x)[/tex]

where [itex]f_n(x)=u_m(x)-u_{m-1}(x)[/itex]

then:

[tex]\int_0^1 \sum_{n=1}^{\infty}f_n(x) dx\neq \sum_{n=1}^{\infty} \int_0^1 f_n(x)dx[/tex]

- #9

WannabeNewton

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