Almost Harmonic

The harmonic series is the divergent infinite series $\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots.$ 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 $\frac{1}{\ldots2018\ldots}\Big):$ $\sum _{n=1, \text{ w/o 2018}}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots+{\frac {1}{2017}}+{\frac {1}{2019}}+\cdots+{\frac {1}{12017}}+{\frac {1}{12019}}+\cdots.$ Which of the following describes the behavior of this almost harmonic series?