TITLE: The series of reciprocals of the integers that do not contain 9 in their decimal representation
QUESTION [5 upvotes]: Does the following series converge or diverge? 
$\sum_{n=1}^{\infty} a_n$
where 
$a_n = \frac 1 b_n$, and
$(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal notation.
I have no idea how the sequence $b_n$ even looks like. If I have an idea of how to prove that a series converges or not with criteria but I don't if I don't know what $b_n$ is.

REPLY [10 votes]: Let $c_n$ be the sum of all reciprocals of $n$-digit positive numbers that contain no '9' digit.  There are $8 \cdot 9^{n-1}$ $n$-digit numbers that contain no '9' digit.  The largest term is the first one, $\dfrac{1}{10^{n-1}}$.  So $c_n < 8 \cdot 9^{n-1} \dfrac{1}{10^{n-1}} = 8 \left(\dfrac{9}{10}\right)^{n-1}$.
You're looking for $\sum c_n$, which is smaller than the geometric series $\sum 8 \left(\dfrac{9}{10}\right)^{n-1}$, so $c_n$ certainly converges.