Convergence of Sum of Reciprocal Palindromes

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.