Get minimum N in approximation of pi

Question

The following expression is an approximation of PI, where N determines the precision.

$$\pi (N) = \frac{4}{N} \sum_{i=1}^{N}\frac{1}{1 +\left ( \frac{i -\frac{1}{2}}{N} \right )^{2}}$$

If I want to find an approximation with an error less than a given amount $E$, I would like to find mathematically the N I have to use, or at least some boundaries where that N has to be.

I've tried approximating the error function, but I couldn't get anything precise enough.

Take into account that for the given error $E$ I can obtain $\pi(N)$.

Answer 1

Clearly, this is a Riemann sum for the integral

$$\pi = 4 \int_0^1 dx \frac{1}{1+x^2}$$

The difference between the Riemann sum above and this integral value is about $|f''(\xi)|/(12 N^2)$, where $f(x)=1/(1+x^2)$ and $\xi$ is the point at which $f''(x)$ is a maximum over $[0,1]$. As this value is $2$ at $\xi = 0$, the value of $N$ for a given error $E$ is about

$$N=\sqrt{\frac{1}{6 E}}$$

Answer 2

I cannot offer a mathematical formula, but at least this should give you an idea of how fast (slow) the series converges to pie, a small table with results:

    1    0.0584
    2    0.0208
    3    0.0093
    4    0.0052
    5    0.0033
    6    0.0023
    7    0.0017
    8    0.0013
    9    0.0010
   10    0.0008
   20    0.0002
   50    0.00003
  100    0.000008
 1000    0.00000008
 2000    0.00000002
 5000    0.000000003
10000    0.0000000008