Was developing some notes on calculus for a nephew, derived Taylor series, and just as an example wrote some code for editor KEdit, with 1000 decimal digit precision, to calculate natural log base e to 36 decimal digits. Amazing how well the code worked. Including the code here.

In TeX, Taylor series is

f(x) = \sum_{i=0}^n {(x - x_0)^i \over i!} f^{[i]}(x_0) + R_n(x_0)

with R_n(x_0) as the error term.

To derive the expression for R_n(x_0), just differentiate f(x) with respect to x_0 where then nearly all the terms cancel, simplify, integrate from x_0 to x, and apply the mean value theorem.

The results are, for some s between x_0 and x:

R_n(x_0) = (x - x_0) {(x-s)^n \over n!} f^{[n+1]}(s)

Final results of the code:

e = 2.71828182845904523536028747135266250

The error is less than

2.90 x 10^(-40)

The code is in the language KEdit macro language Kexx, a version of Rexx:

          macro_name = 'NATLOG'

          out_file = macro_name || '.out'

          'nomsg erase' out_file

          Call msgg macro_name':  Find natual logarithm base e'

          numeric digits 1000

          n = 35

          sum = 1

          factorial = 1

          Do i = 1 To n
            factorial = i * factorial
            sum = sum + 1/factorial
            Call msgg Format(i, 5) Format(factorial, 50) Format(sum, 2, 35)
          End

          error = 3 / factorial

          Call msgg macro_name':  The error is <='

          Call msgg Format( error, 59, 50 )

          Call Lineout out_file

          Return
     msgg:
          Procedure expose out_file

          Call Lineout out_file, arg(1)

          Return