I'm writing the notes for
my 9 year old niece.
So, the notes are for a boy of 9.
And it's calculus, for a boy of 9. Of course it's "calculus 1".
I thought I mentioned it was for a boy of 9.
So, I omitted (1) a real valued function of a real variable with a compact domain has a Riemann integral if and only it is continuous on a set of measure zero, (2) for such function, if it is differentiable, then it has no jump discontinuities, (3) there is such a function that is differentiable but whose Riemann integral does not exist.
But, still, it is amazing how easy it is to get 36 decimal digits of e, and 500 if want.
I've been through calculus, advanced calculus, advanced calculus for applications, differential equations, local series solutions to the Navier-Stokes equations, exterior algebra, real analysis, measure theory, and more, have taught calculus in college, applied it in US national security and business, and published in it, and still I'd never seen a clear treatment of how easy it is to get so many digits of e from Taylor series.
The results I found are amazing, and
my good and long experience indicates that only a tiny fraction of calculus students appreciate that.
And as we know,
e = lim_{i \rightarrow \infty}
(1 + 1/i)^i
and it turns out that that iteration is painfully slow, and from my experience this fact is also amazing and poorly known.
