Friday, September 1, 2023

Newton's series for the exponential

I've been reading the first chapter of Dunham's Calculus Gallery again (available here).

It starts with a discussion of Newton realizing that the binomial theorem (a+b)^n also applies for rational r as in (a+b)^r. In that case, the series does not terminate but is infinite. There is a deep discussion of how he came to that by Dennis and Addington, here.

Many great things come out of that including series for the logarithm and inverse sine and several series for π. Dunham also illustrates how Newton inverted or reversed series, for example to turn the inverse sine into the sine and the logarithm into the exponential.

I have new write-ups posted on github including an introduction to the standard binomial (pdf), as well as a second one working through the examples of of Dunham's chapter (pdf), including the process of inverting series, and a derivation of the exponential, as Newton did it.

Dunham left that to the reader. I haven't been this excited since Gil Strang led me to the integral that Newton solved, showing that for an inverse square force the mass acts as a point mass. To see it through Newton's eyes is a rare treat.

I found versions of Newton's manuscripts online (some in Latin), but haven't yet located the material on the binomial and on series.

I know I would have been very excited to find my second chapter linked here. Since visitors here have dropped from thousands to double digits, one can only hope that someone will click through and be excited as well.

Many thanks to a reader on math.stackexchange for pointing out my elementary error in a nice way, which allowed me to finish the last bit.