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Prove that the sum of the reciprocals of all positive integers which do not contain the digit $9$ converges.

I can't see an easy way to write out what the series is explicitly, so I think I will have to compare it to another series which we know converges and is similar.

Puzzled417
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  • To be pedantic: interpreted literally, the sum is vacuously zero, since there are no integers that contain the digit 9. Perhaps you mean the positive integers whose decimal representations contain the digit 9 ;) – symplectomorphic Jun 09 '16 at 04:17
  • I have not solved this, as they said in the Italian Job, I have an idea... How many integers are knocked out by this rule - consider the ranges 1-9, 10-99, 100-999, 1000-9999, etc. What is the largest reciprocal value for each of these ranges? What is the upper bound on the sum of these reciprocals for each range? You should be able to form an upper bound based up on terms of a geometric series which converges. Interesting problem. – user247608 Jun 09 '16 at 04:30

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