scipy.optimize.minimize and Minuit returning initial guess value

scipy.optimize.minimize is primarily used for continuous differentiable functions. Using arange in func makes a discrete problem from it. This causes big jumps in the gradient due these discontinuities (see the picture below).

I added some debug print:

from time import process_time
import numpy as np
from scipy.optimize import minimize
class NMin(object):
    def __init__(self, error):
        self.error=error

    def func(self, N):
        print("func called N = {}".format(N))
        i = np.arange(1, N+1)
        print("i = {}".format(i))
        f = np.abs(np.sum(4/(N*(1+((i - 0.5)/N)**2))) - np.pi)-self.error
        print("f = {}".format(f))
        return(f)

    def nMin(self):
        x0 = 1
        nMin = minimize(self.func, x0)
        return(nMin.x)


def main():
    t1_start = process_time()
    error=10**(-6)
    nMin = NMin(error).nMin()
    print("the minimum value of N is: " + str(nMin))
    t1_stop = process_time()
    print("Elapsed time during the whole program in seconds:",
                                         t1_stop-t1_start)

main()

This results in:

func called N = [1.]
i = [1.]
f = 0.05840634641020706
func called N = [1.00000001]
i = [1. 2.]
f = 1.289175555623012

Maybe you would like to use a different solver more suitable for discrete problems or change your objective to satisfy the prerequisite of continuity for gradient-based optimization.