In yesterday’s post Not-so Easy e, I briefly covered The Taylor Series and how it is applied to solving for the value of , or for specifically finding the value of e. I didn’t spend a lot of time covering The Taylor Series, and skipped some steps on my path to deriving the value of e. I figured I should take some time to go back over Taylor, show why it was chosen, and how it fits in with e.

The Taylor Series has a rich history in mathematics, and it is used to solve a number of classic problems. The Taylor Series is defined in Taylor’s Theorem. If a function (we used yesterday) is differentiable n times on a closed interval [a,x] and n + 1 times over the open interval (a,x), and , the function can be precisely approximated by The Taylor Series.

Math note: The open interval (a,x) includes all values between a and x exclusively, and the closed interval [a,x] includes all values between a and x inclusively.

I am not going to go through the whole theorem here, but I will show in more detail how it is applied to . Lets go through step by step.

#01 – Original equation:

#02 – First Derivative:

#03 – Taylor:

#04 – What value a should we use that is near x? We will use 0, which will produce a value of 1 when the function .

#05 – Substitute in derivatives:

#06 – Obvious:

#07 – Finished:

I have stumbled over uses for The Taylor Series from time to time working as a software engineer. Its pretty easy to compute a value to an arbitrary precision using a loop. In the practical world an approximation is good enough, and even a loop that is run 100 times with most math should return almost immediately on modern hardware. Apparently this is how scientific calculators do a number of computations. Very cool.

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