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.

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.

anonymous · Answer

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.

anonymous · Answer

I meant $n!-2$, but ok.