jmc

The first addend, $1.11111111\dots$, is by itself equal to the sum of the infinite geometric series $$1+\frac{1}{10}+{\left(\frac{1}{10}\right)}^2+{\left(\frac{1}{10}\right)}^3+\cdots ,$$which is $\frac{1}{1-\frac{1}{10}}=\frac{10}{9}$.

The second addend is one-tenth of this, so equals $\frac{1}{9}$. The third addend is one-tenth of the second addend, and so on. Thus the sum of the infinite column of infinite decimals is $$\begin{array}{rl}\frac{10}{9}\cdot \left[1+\frac{1}{10}+{\left(\frac{1}{10}\right)}^2+\cdots \right]& =\frac{10}{9}\cdot \frac{10}{9}\\ & =\boxed{\frac{100}{81}}.\end{array}$$Notice that we have just added $1+\frac{2}{10}+\frac{3}{100}+\frac{4}{1000}+\cdots$, sneakily.