data/problems/368.yml
---
:id: 368
:name: A Kempner-like series
:url: https://projecteuler.net/problem=368
:content: "The **harmonic series** $1 + \\dfrac{1}{2} + \\dfrac{1}{3} + \\dfrac{1}{4}
+ ...$ is well known to be divergent.\n\nIf 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. \nThis modified harmonic series is called the **Kempner**
series.\n\nLet 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. \nThese 20 omitted terms are:\n\n$$\\dfrac{1}{111},
\\dfrac{1}{222}, \\dfrac{1}{333}, \\dfrac{1}{444}, \\dfrac{1}{555}, \\dfrac{1}{666},
\\dfrac{1}{777}, \\dfrac{1}{888}, \\dfrac{1}{999}, \\dfrac{1}{1000}, \\dfrac{1}{1110},
\\\\\\ \\dfrac{1}{1111}, \\dfrac{1}{1112}, \\dfrac{1}{1113}, \\dfrac{1}{1114}, \\dfrac{1}{1115},
\\dfrac{1}{1116}, \\dfrac{1}{1117}, \\dfrac{1}{1118}, \\dfrac{1}{1119}$$\n\nThis
series converges as well.\n\nFind the value the series converges to. \nGive your
answer rounded to 10 digits behind the decimal point.\n\n"