Home » Blog » Prove that the sum of reciprocals of integers with no zeros in their decimal representation converges

# Prove that the sum of reciprocals of integers with no zeros in their decimal representation converges

Consider the positive integer with no zeros in their decimal representation: Prove that the series converges. Further, prove that the sum is less than 90.

Incomplete.

1. Rafael Deiga says:

Solution:
First, note that 1/b<(1/10)^(k-1) for all b with k digits (and without the digit 0). Furthermore, there are exactly 9^k numbers b with k digits, because there are nine possibilites for each digit. Therefore, the sum of all b's is smaller than 9*(9/10)^(k-1). Thus, 1+1/2+1/3+1/4+… < 9*(1+9/10+(9/10)^2+…), which converges to 90.

Sorry about my solution, I don't know how to use latex here.

• Thank you!!!

• Anonymous says:

This is very good. But there is a small mistake, 1/b<(1/10)^(k-1) is not true for b = 1, k=1. But if you change it to <= instead of <. Unfortunately this does not guarantee that the sum is less than 90 only that the sum is less than or equal to 90. To prove that it is less than 90 write out the first ten terms and sum them up and add the rest with the geometric series formula.