Reciprocal Sum of Palindromes

A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n=(a_k{a}_{k-1}\cdots a_0{)}_b$ in base $b$ with $a_k\mathrm{\ne }0$ has the symmetric property $a_{k-i}=a_i$ for every $i=0,1,2,\dots ,k$. Let $s_b$ be the reciprocal sum of all $b$-adic palindromes. It is not difficult to show that $s_b$ converges. In this article, we obtain upper and lower bounds for $s_b$ and the inequality $s_b<s_{b^{\mathrm{\prime }}}$ for $2\le b<b^{\mathrm{\prime }}$. Its consequences and some numerical data are also given.