In [1]: 
from __future__ import division
In [2]: 
n = int(0)

float_type = np.float32

my_sum = float_type(0)

while True:
    n += 1
    last_sum = my_sum
    my_sum += float_type(1 / n)
    
    if n % 200000 == 0:
        print "1/n = %g, sum0 = %g"%(1.0/n, my_sum)
        
    if last_sum == my_sum:
        print "Stopped changing at %d" % n
        break

1/n = 5e-06, sum0 = 12.7828
1/n = 2.5e-06, sum0 = 13.4814
1/n = 1.66667e-06, sum0 = 13.8814
1/n = 1.25e-06, sum0 = 14.1666
1/n = 1e-06, sum0 = 14.3574
1/n = 8.33333e-07, sum0 = 14.5481
1/n = 7.14286e-07, sum0 = 14.7388
1/n = 6.25e-07, sum0 = 14.9296
1/n = 5.55556e-07, sum0 = 15.1203
1/n = 5e-07, sum0 = 15.311
Stopped changing at 2097152
In [2]: