Expressing the infinite sum $1 + 22 + 333 + 4444 + \dotsb$

The terms of the sequence are

$$a_n =\left(n-10\cdot\left\lfloor\frac{n}{10}\right\rfloor\right)\cdot\frac{10^{n}-1}{9}$$

the sum of your series is $\infty$

$$\sum_{n=1}^{\infty}\left(n-10\cdot\left\lfloor\frac{n}{10}\right\rfloor\right)\cdot\frac{10^{n}-1}{9}$$