A "large" subset of positive
integers is one whose sum-of-reciprocals diverges, whereas a "small" subset of
positive integers has a convergent sum-of-reciprocals. The odd positive
integers, for example, is a "large" set. The squares, on the other hand,
is "small".

What about the numbers, base r, that can be written without using one
particular digit, d? Is this set large or small? Answer: the set is
small -- the set of reciprocals of these numbers converges.

### The sum of reciprocals of numbers that can be written without using digit d,
base r, converges.

Let S_{1} be the sum of reciprocals of one-digit numbers, base r,
that can be written without using digit d.

Let S_{2} be the sum of reciprocals of two-digit numbers, base r,
that can be written without using digit d.

etc.

S_{1} is the sum of r-2 reciprocals, none bigger than 1, so S_{1}
≤ r-2

S_{2} is the sum of (r-2)(r-1) reciprocals, none bigger than 1/r, so
S_{2} ≤ (r-2)(^{r-1}/_{r})

S_{3} is the sum of (r-2)(r-1)^{2} reciprocals, none bigger
than 1/r^{2}, so S_{3} ≤ (r-2)(^{r-1}/_{r})^{2}

etc.

So S_{1} + S_{2} + S_{3} + ... ≤ (r-2)( 1 + (^{r-1}/_{r})
+ (^{r-1}/_{r})^{2}
+ ... ) = (r-2)(r)

### Facts about Small and Large sets

A union of any finite number of small sets is a small set.

A subset of a small set is a small set.

The compliment of a small set in a large set is large. That is, if L is
a large set, and S is a small set, then the set L-S is large, where L-S consists
of all elements of L that are not in S.

Any large set can be partitioned into two (or any finite number of) large
sets, all of which are large. (One method is to assign elements of L in
order to L_{1}, L_{2}, ..., L_{n}, L_{1}, L_{2},
..., L_{n}, in "round robin" fashion).

The primes are large.

### Related pages in this website

Sets

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