1. ## Pi as summation

So I've seen that pi can be written as a series:

$\displaystyle pi = 4 * series(((-1)^k)/2k+1)$

So pi = 4/1 - 4/3 + 4/5 - 4/7 + ...

Now I know that Q is an ordered field so if a and b are in Q, if I add them, the result a+b is another element in Q.

But if I sum up infinitely many rational numbers I can get an irrational? Wondering how this works.

2. Originally Posted by anomaly
So I've seen that pi can be written as a series:

$\displaystyle pi = 4 * series(((-1)^k)/2k+1)$

So pi = 4/1 - 4/3 + 4/5 - 4/7 + ...

Now I know that Q is an ordered field so if a and b are in Q, if I add them, the result a+b is another element in Q.

But if I sum up infinitely many rational numbers I can get an irrational? Wondering how this works.
Is this really that surprising? Isn't every irrational number the limit of a sequence of rational approximations, which can be represented as the infinite sum of rationa numbers.

3. I know, but when you take the limit of a sum of a bunch of rationals you get an irrational, even though the rationals are closed under addition? That's what's strange about it.

4. Originally Posted by anomaly
I know, but when you take the limit of a sum of a bunch of rationals you get an irrational, even though the rationals are closed under addition? That's what's strange about it.