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Math Help - a series problem

  1. #1
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    a series problem

    Consider the sum \frac{1}{1}+\frac{1}{2}+...+\frac{1}{8}+\frac{1}{1  0}+...+\frac{1}{18}+\frac{1}{20}+...+\frac{1}{88}+  \frac{1}{100}+... where we sum over all \frac{1}{n} where n 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 n has exactly k 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?
    Last edited by DontKnoMaff; October 16th 2010 at 05:53 PM.
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  2. #2
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    Quote Originally Posted by DontKnoMaff View Post
    Consider the sum \frac{1}{1}+\frac{1}{2}+...+\frac{1}{8}+\frac{1}{1  0}+...+\frac{1}{18}+\frac{1}{20}+...+\frac{1}{88}+  \frac{1}{100}+... where we sum over all \frac{1}{n} where n 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 n has exactly k 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
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