In Madhava Mathematics Competition 2015 (held in January 2015), we were asked to prove the convergence of Kempner Series (first time proved in 1914). Recently I discovered the paper by A. J. Kempner (http://www.jstor.org/stable/2972074), so in this blog post I will state and prove that problem.
The basic idea behind proof is to divide whole series into chunks (finding symmetry) and then construct a converging geometric series which will act as upper bound of Kempner Seires.
Theorem (Kempner Series, 1914). Harmonic Series,, converge, if the denominators do not include all natural numbers , but only those numbers which do not contain any figure 9.
Proof: Given Series is:
Now we can rewrite above series as:
where is the sum of all terms in of denominator with .
Observe that, each term of which forms part of , is less than or equal to .
Now count the number of terms of which are contained in , in , , in . Clearly, , consists of 8 terms, and . In there are, as is easily seen, less than terms of , and . Altogether there are in less than terms with denominators under 100.
Assume now that we know the number of terms in which are contained in to be less than , for . Then, because each term of which is contained in is not greater than , we have , and the total number of terms in with denominators under is less than .
Now, let’s go for induction. For and we have verified all this, and we will now show that if it is true for , then . contains all terms in of denominator , . This interval for can be broken up into the nine intervals, , . The last interval does not contribute any term to , the eight remaining intervals contribute each the same number of terms to , and this is the same as the number of terms contributed by the whole interval , that is, by assumption, less than .
Therefore, contains less than terms of , and each of these terms is less than or equal to , we have .
Thus, converges, and since, , also converges.
Note: There is nothing special about 9 here, the above method of proof holds unchanged if, instead of 9, any other figure is excluded, but not for the figure .