Question

Prove mathematically by hand that\[\sum_{n=1}^\infty10^{-n!}\]evaluates to a number with a 1 in each decimal place corresponding to a factorial, and 0 every else.

Not sure if this is doable.

Answer 1

Oh, well, predictably\[10^{-n!}=0.0\underbrace\ldots_{n-2}01\]Therefore additional sums will add additional terms to the end.

Whoops, easier question than I expected.

Answer 2

I meant \(n!-2\), but ok.