Can the starting point of an infinite series be a non-integer?

Question

Can one have for example $$\sum_{k=\pi}^{\infty}\frac{1}{k^2}$$

The idea being that one would successively plug in $\pi$, then $\pi+1$, then $\pi+2$ and so on.

If it is possible, there are surprising (for me) consquences. Take the function q(x), that is 1 if x rational and 0 otherwise. Then the sum

$$\sum_{k=1}^{\infty}q(k)$$ diverges, but

$$\sum_{k=\pi}^{\infty}q(k)$$ converges.

So is this possible?

Specifically, I am dealing with $\sum_{k=1}^{\infty}\frac{|sin(k)|}{k}$ and I wanted to make the transformation $k = \pi/2(2z+1)$, but then the new series starts at $1/\pi-1/2$

Answer 1

Yes, you may, but keep in mind that it is not useful. I have never seen a sum tht uses irrational numbers, and that's for a good reason. You won't be able to do anything useful.

In answer to your "weird" case, it is only an result of 1) the fact that sum notation goes stepwise in integers, and 2) the function you chose.

Hope this helps!

Answer 2

Sure, if you have any ordered set of indices $\mathcal{A}$ and a collection of numbers $\{\rho_\alpha \}_{\alpha\in \mathcal{A}}$, you can write $$ \sum_{\alpha\in\mathcal{A}}\rho_\alpha $$