Tricky harmonic sum without some terms

Asked Jan 21 '16 at 13:51

Active Jan 21 '16 at 13:57

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I have to prove that the harmonic sum: $$\sum_{n=1}^\infty \frac1n$$

converges when we remove all the terms $\frac1n$ where the decimal expansion of $n$ contains the digit $9$.

I have a hint but don't know how to use it; how many terms of at most $k$ digits are there?

edited Jan 21 '16 at 13:57

Alex M.

asked Jan 21 '16 at 13:51

Ergo

https://en.wikipedia.org/wiki/Kempner_series – RFZ Jan 21 '16 at 13:53
4

There are many question on math.SE covering this sort of sum. Look at answers of this question and the "Linked" questions there. – achille hui Jan 21 '16 at 13:56

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