You're probably already aware that the harmonic series,
which is the sum of
the reciprocals of all natural numbers, diverges.
In fact, it diverges if you take away every other term.
It even diverges if you take away nine out of every
So, what do you think would happen if we tried to take
the sum of reciprocals of all natural numbers
that do not contain the number nine
(when written in decimal expansion)?
Amazingly, this series converges!
Write out the first several terms.
Allow students to guess whether or not this converges.
For instance, it may appear that this series is divergent,
especially when contrasting it with sums of reciprocals of
numbers with one or more 9's.
That series diverges (easy to show), and this series seems
to have "more" terms in it...
The Math Behind the Fact:
Group the terms based on the number of digits
in their denominator.
There are 8 terms in (1/1+...+ 1/8) each of which
is no larger than 1.
Consider the next group (1/10+...+1/88). The number of
terms is at most the number of ways to choose
two ordered digits out of the digits 0..8, and each such
term is clearly no larger than 1/10.
So this group's sum is no larger than 92/10.
Similarly, the sum of the terms in
(1/100+...+1/999) is at most
So the entire sum is no larger than
9*1 + 9*(9/10) + 9*(92/102) +
... + 9*(9n/10n) + ...
This a geometric series that converges. Thus by
the comparison test, the original sum (which is
smaller term-by-term) must converge!
How to Cite this Page:
Su, Francis E., et al. "Thinned-Out Harmonic Series."
Math Fun Facts.