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 = (aₖ aₖ₋₁ ⋯ a₀)b in base b with aₖ ≠ 0 has the symmetric property aₖ₋ᵢ = aᵢ for every i=0,1,2,… ,k. Let sb be the reciprocal sum of all b-adic palindromes. It is not difficult to show that sb converges. In this article, we obtain upper and lower bounds for sb and the inequality sb <sb' for 2≤ b<b'. Its consequences and some numerical data are also given.