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.**

Skip to content
#
Stumbling Robot

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

###
2 comments

### Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):

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.**

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!!!