Almost Harmonic

Calculus Level 3

The harmonic series is the divergent infinite series n=11n=1+12+13+14+15+.\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots. Define the almost harmonic series as the sum of the inverses of all the numbers that don't contain the string 2018 (\Big(i.e. all the terms of the form 12018):\frac{1}{\ldots2018\ldots}\Big): n=1, w/o 20181n=1+12+13++12017+12019++112017+112019+.\sum _{n=1, \text{ w/o 2018}}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots+{\frac {1}{2017}}+{\frac {1}{2019}}+\cdots+{\frac {1}{12017}}+{\frac {1}{12019}}+\cdots. Which of the following describes the behavior of this almost harmonic series?

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