Calculating Napier constant (e) using recursion

Question

I am attempting to approximate the value of e (~2.7)

Defined by this, for each nth term

using a recursive function in python.

So far I have gotten this,

def NapierConstant(runs):
    return 2 + 1/contfrac(1, 2, runs)

def contfrac(v1, v2, limit):
    if v1 == limit:
        return (v1/v2)
    else:
        return v1+(v1/contfrac(v1+1, v2+1, limit))

print(NapierConstant(2))

this should output 2.72727 here, but instead I get 2.4, and the error margin get's worse for each following step. I have googled, and I can't manage to figure out how to set up the function recursively so that it outputs the expected values.

Answer 1

def get_e(lim):
    return 2 + 1/r(1, lim)

def r(v1, lim):
    if v1 == lim:
        return v1 + v1/(v1+1)
    else:
        return v1 + v1/(r(v1+1, lim))