Convergence of Sum of Reciprocal Palindromes Let $P_{b}(n)$ be the $n$th palindromic number in base $b$. I read on Wikipedia that the sum
$$\sum_{n=1}^\infty \frac{1}{P_{10}(n)}$$
converges, but I have no idea how to prove it.
I tried using combinatorics to count the "density" of the palindromes and establish an upper bound that was easy to prove was convergent, but my counting just got too complicated. How do I prove that this converges? Furthermore, how can I prove (or disprove) that it converges for any $b\ge 2, b\in \mathbb N$?

For numbers with $k$ digits there are $10^{k/2}$ palindromes if $k$ is even and $10^{(k+1)/2}$ if $k$ is odd.  The reciprocal of such a palindrome is less than $10^{-(k-1)}$.  The sum of all the reciprocal palindromes of $k$ digits is less than $10^{-k/2}$.  The sum over $k$ is a geometric series with ratio less than $1$, so the sum converges.  The same argument works in any base.
