How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out how many terms I need (or, imagine that number of decimal places is high enough that trial and error is not efficient)? The problem is that equating the error (e.g. 5*10^(-13) in my example above) to the integral form of the remainder leaves us an expression that can be numerically (but certainly not analytically) solved, whereas I am looking for a method that works (even approximately) by hand and handheld calculator.