Newton-Raphson not working - future value annuity due formula

Question

I'm coding a 'Future Value of Annuity Due' calculator, that allows a user to find an unknown in the formula. The formula is fv = (1 + i) * pp * ((1 + i)**n - 1) / i; where fv is future value, pp is periodic payment, i is interest rate and n is number of periods. As an example, assuming pp = 100, i = .2735 (27.35%), and n = 11, one obtains fv = 6187.56. Without knowing i, I can interpolate the rate to be say 25%, and would like to use Newton-Raphson iteration to get a more exact answer. However, my code below is off, as it's diverging (it seems to work for small values of i, i.e. 5%).

fv = 11807.795
pp = 1000
n = 10
i = .03

def newton_raphson_method(fv,pp,i,n):
    newton_raphson_i = i
    for num in range(1,20):
        newton_raphson_i = i - (1+i)*(pp*(1+i)**n - pp-fv*i) / ((n +1)*pp*(1+i)**n - fv)
        i = newton_raphson_i
        print(i)
    i = round(i,11)
    print('')
    print ('The newton interest rate is ' + str("%.9f" % (i * 100)) + '%')
    print('')

Answer 1

For the Future Value of an Annuity Due Formula, the above formulas were both missing the periodic payment (pp) at the end. The below works with positive and negative interest rates.

newton_raphson_i = i - ((1+i)*pp*((1+i)**n - 1) - fv*i) / ((n+1)*pp*(1+i)**n - fv - pp)