## A Kempner-like series

### Problem 368

Published on Sunday, 22nd January 2012, 01:00 am; Solved by 358; Difficulty rating: 45%The ** harmonic series ** $1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + ...$ is well known to be divergent.

If we however omit from this series every term where the denominator has a 9 in it, the series remarkably enough converges to approximately 22.9206766193.

This modified harmonic series is called the **Kempner** series.

Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has 3 or more equal consecutive digits.
One can verify that out of the first 1200 terms of the harmonic series, only 20 terms will be omitted.

These 20 omitted terms are:

$$\dfrac{1}{111}, \dfrac{1}{222}, \dfrac{1}{333}, \dfrac{1}{444}, \dfrac{1}{555}, \dfrac{1}{666}, \dfrac{1}{777}, \dfrac{1}{888}, \dfrac{1}{999}, \dfrac{1}{1000}, \dfrac{1}{1110}, \\\ \dfrac{1}{1111}, \dfrac{1}{1112}, \dfrac{1}{1113}, \dfrac{1}{1114}, \dfrac{1}{1115}, \dfrac{1}{1116}, \dfrac{1}{1117}, \dfrac{1}{1118}, \dfrac{1}{1119}$$

This series converges as well.

Find the value the series converges to.

Give your answer rounded to 10 digits behind the decimal point.