Thread: a series problem

Consider the sum where we sum over all where doesn't have the digit 9 in its decimal expansion. Show that the series converges. Hint: How many terms are there in the sum where has exactly digits?

My Work:
I see that the hint is 9^k-1 since every digit can be anything from 0-8 but not all digits can be 0. I feel like an epsilon argument might work here? But how will the number of terms help me?

Originally Posted by DontKnoMaff

Consider the sum where we sum over all where doesn't have the digit 9 in its decimal expansion. Show that the series converges. Hint: How many terms are there in the sum where has exactly digits?

My Work:
I see that the hint is 9^k-1 since every digit can be anything from 0-8 but not all digits can be 0. The question seems a little confusing since it talks about the sum and then asks if the series converges.. I feel like an epsilon argument might work here? But how will the number of terms help me?

What you have is a depleted harmonic series. See here: Nick's Mathematical Puzzles: Solution 72