# How to Integrate a Sigma Sum?

1. Oct 21, 2013

### Superposed_Cat

How would you Integrate a Sigma Sum?

2. Oct 21, 2013

### arildno

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

3. Oct 21, 2013

### Staff: Mentor

I.e., term by term.

4. Oct 21, 2013

### Superposed_Cat

thanks.

5. Oct 21, 2013

### arildno

Not necessarily, but usually beneficially!

6. Oct 21, 2013

### 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.

7. Oct 21, 2013

### Superposed_Cat

what?

8. Oct 21, 2013

### jackmell

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:

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

9. Oct 21, 2013

### 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.