# Pi as summation

#### 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.

#### Drexel28

MHF Hall of Honor
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.

#### 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.

#### Drexel28

MHF Hall of Honor
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.

• anomaly

#### anomaly

So when you take a limit of something you can converge to something completely different?

#### Drexel28

MHF Hall of Honor
So when you take a limit of something you can converge to something completely different?
I have no real idea what that means, but I suppose the answer is yes.

#### anomaly

I mean having a summation converging to a value that is not within the original set.

Ok, thank you.