Can a Sigma Sum be Integrated with Fubini's Theorem?

[Superposed_Cat]

How would you Integrate a Sigma Sum?

 

[arildno]

If you with a sigma sum means a sum written with the big sigma symbol, you integrate it just like any other sum.

 

[Mark44]

If you with a sigma sum means a sum written with the big sigma symbol, you integrate it just like any other sum.

I.e., term by term.

 

[Superposed_Cat]

thanks.

 

[arildno]

I.e., term by term.

Not necessarily, but usually beneficially!

 

[FeDeX_LaTeX]

On a related note, an interesting question to consider would be when the orders of a sum and an integral can be interchanged.

 

[Superposed_Cat]

what?

 

[jackmell]

On a related note, an interesting question to consider would be when the orders of a sum and an integral can be interchanged.

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: $u_m(x)=mxe^{-mx^2}$

and consider:

$$\sum_{n=1}^{\infty} f_n(x)$$

where $f_n(x)=u_m(x)-u_{m-1}(x)$

then:

$$\int_0^1 \sum_{n=1}^{\infty}f_n(x) dx\neq \sum_{n=1}^{\infty} \int_0^1 f_n(x)dx$$

 

[WannabeNewton]

Of course you have to make sure the sum and integral are actually interchangeable because this is not always the case. The sufficient condition is a special case of Fubini's theorem.