def L(x, n):
    s = 0
    for i in range(1, n+1):
        s += (1.0/i)*(x/(1.0+x))**i
    return s

def L2(x, n):
    s = 0
    for i in range(1, n+1):
        s += (1.0/i)*(x/(1.0+x))**i
    value_of_sum = s
    first_neglected_term = (1.0/(n+1))*(x/(1.0+x))**(n+1)
    from math import log
    exact_error = log(1+x) - value_of_sum
    return value_of_sum, first_neglected_term, exact_error


def table(x):
    from math import log
    print('\nx=%g, ln(1+x)=%g' % (x, log(1+x)))
    for n in [1, 2, 10, 100, 500]:
        value, next, error = L2(x, n)
        print('n=%-4d %-10g  (next term: %8.2e  '\
              'error: %8.2e)' % (n, value, next, error))

table(10)
table(100)
table(1000)

def L3(x, epsilon=1.0E-6):
    x = float(x)
    i = 1
    term = (1.0/i)*(x/(1+x))**i
    s = term
    while abs(term) > epsilon:
        i += 1
        term = (1.0/i)*(x/(1+x))**i
        s += term
    return s, i

def table2(x):
    from math import log
    for k in range(4, 14, 2):
        epsilon = 10**(-k)
        approx, n = L3(x, epsilon=epsilon)
        exact = log(1+x)
        exact_error = exact - approx
        print('epsilon: %5.0e, exact error: %8.2e, n=%d' % \
              (epsilon, exact_error, n))

print('\n\n')
table2(10)