An easier way to evaluate sum i/10^i is by squaring sum 1/10^i

If you multiply term by term every term has coefficient 1 of course. There are n terms with exponent n+1, made from the n sums of the first exponent and the second exponent.

Eg 1+5, 2+4, 3+3, 4+2, 5+1.

So (1/9)^2 = (sum 1/10^i)^2 = 1/10 sum i/10^i

The derivative trick is more useful generally, but this method gets you the solution to 0.12345678.. in an quick way that's also easier to justify that it works.