# How to Integrate a Sigma Sum?

Superposed_Cat
How would you Integrate a Sigma Sum?

Homework Helper
Gold Member
Dearly Missed
If you with a sigma sum means a sum written with the big sigma symbol, you integrate it just like any other sum.

Mentor
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.

Homework Helper
Gold Member
Dearly Missed
I.e., term by term.
Not necessarily, but usually beneficially!

Gold Member
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:

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