Room isn't enough for 2

Calculus Level 1

Consider the sum of the reciprocals of every positive integer that doesn't contain the digit 2:

\[\frac{1}{1} + \frac{1}{3} + \frac 14 + \cdots + \frac{1}{10} + \frac{1}{11} + \frac{1}{13} +\cdots+ \frac{1}{19} + \frac{1}{30} + \frac 1{31} \cdots.\]

Does this sum converge?

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