Integral of a measurable function over N

Question

I want to calculate the integral of the function $a:\mathbb{N}\to\mathbb{R}$, $a(n)= \frac{(-1)^n}{2^n}$ over the set $\mathbb{N}$, with the cardinal measurement.

I've seen how to calculate this type of integrals on simple functions, but this is not a simple function, and I can't find any examples on how to calculate this type of integrals.

Any kind of help will be much appreciated, thank you.

Answer 1

Note that the integration with respect to the counting measure is just the (infinite) sum, see for example here for a proof. So the the desired integral is $$\int_{\Bbb N}a~\mathrm{d}\mu=\sum_{n=1}^\infty \frac{(-1)^n}{2^n}=\cdots$$

Edit: the proof in the link only shows the result for non-negative bounded functions. But of course our case follows from that by splitting it up into a negative and positive part.