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
.
Hence, 
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
.
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