*e*, found by using more and more terms of its Taylor's series. As we add terms, the approximation improves for small |

^{x}*x*| and diverges more slowly for large |

*x*|. Taking additional terms from the Taylor's series approximates the exponential by higher and higher order polynomials. This is all interesting and useful, but the exponential looks similar to a polynomial, at least for positive

*x*, so it is not too surprising that polynomials do a decent job approximating the exponential.

A more challenging function to fit with a Taylor’s Series would look nothing like a polynomial, which always grows to plus or minus infinity at large |

*x*|. I wonder how the Taylor’s Series does approximating a bounded function; perhaps a function that oscillates back and forth a lot? The natural choice is the sine function.

The Taylor’s Series of sin(

*x*) is

sin(

*x*) =*x*–*x*^{3}/6 +*x*^{5}/120 –*x*^{7}/5040 +*x*^{9}/362880 - …The figure below shows the sine function and its various polynomial approximations.

The red curve is the sine function itself. The simplest approximation is simply sin(

*x*) =

*x*, which gives the yellow straight line. It looks good for |

*x*| less than one, but quickly diverges from sine at large |

*x*|. The green curve is sin(

*x*) =

*x*–

*x*

^{3}/6. It rises to a maximum and then falls, much like sin(

*x*), but heads off to plus or minus infinity relatively quickly. The cyan curve is sin(

*x*) = x –

*x*

^{3}/6 +

*x*

^{5}/120. It captures the first peak of the oscillation well, but then fails. The royal blue curve is sin(

*x*) =

*x*–

*x*

^{3}/6 +

*x*

^{5}/120 –

*x*

^{7}/5040. It is an excellent approximation of sin(

*x*) out to

*x*= π. The violet curve is sin(

*x*) =

*x*–

*x*

^{3}/6 +

*x*

^{5}/120 –

*x*

^{7}/5040 +

*x*

^{9}/362880. It begins to capture the second oscillation, but then diverges. You can see the Taylor’s Series is working hard to represent the sine function, but it is not easy.

Appendix D in IPMB gives a table of values of the exponential and its different Taylor’s Series approximations. Below I create a similar table for the sine. Because the sine and all its series approximations are odd functions, I only consider positive values of

*x*.

One final thought. Russ and I title Appendix D as "Taylor's Series" with an apostrophe s. Should we write "Taylor Series" instead, without the apostrophe s? Wikipedia just calls it the "Taylor Series". I have seen it both ways, and I don't know which is correct. Any opinions?

I vote for "Taylor Series." In Bill Bialek's recent biophysics book there's a nice footnote on Reynolds number, Bessel functions, and other apostrophe-less things: https://books.google.com/books?id=5In_FKA2rmUC&lpg=PA154&dq=bialek%20reynolds%20number&pg=PA154#v=onepage&q&f=false

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