Note: all summations in this answer are implied to range from n=0 to n=∞, with Δn set to equal 1.
The error function is an integral of the bell curve between 0 and x, as we all know. The bell curve is equal to e^(-x^2)*2/sqrt(π).
e^x= Σ (x^n)/(n!)
e^(-x^2)= Σ ((-x^2)^n)/(n!)
e^(-x^2)2/sqrt(π)=2/sqrt(π) Σ ((-x^2)^n)/(n!)
To integrate this function, we simply multiply each term in the polynomial this summation represents by x/(exponent in term+1). The exponent in each term is 2*n, so we multiply each term by x/(n+1). Since x is consistent throughout the summation, though, we can just multiply the entire summation by x and divide each term by (2n+1).
erf(x)= 2x/sqrt(π)* Σ ((-x^2)^n)/(n!*(2n+1))
And there you have it, an exact representation of the error function. If you don't believe me, test it out on a graphing calculator. You'll notice for each term you add (n=0, n=1, n=2), your graph will get closer and closer to the error function.