Modifications of the Harmonic Series - Department of Mathematics

We can also use these theorems for modifications of the harmonic series in
different bases. Although the series will still be partitioned based on the number of
digits in the denominator, the values of and
i
However, if b is the base, then
N i
M will change as the base changes.
i 1
i = ( −1) ⋅ and we can choose ( ) 1
1 i−
M i b
N b b −
= .
Then we still have that M i+ 1Ni+ 1 MiNi − ∑ converges, and { MiN i}
is monotonically
decreasing. Table 4 shows a few modifications of the harmonic series in different
bases.
Table 4
Base Modification** i r i p
1
3 The digit 2 is forbidden 12 i−
1
4 The digit 2 is forbidden 23 i−
1
7 The digit 2 is forbidden 56 i−
1
12 The digit 2 is forbidden 10 11 i−
⋅ ( ) 1
1 2
2 3
i−
⋅ ( ) 1
2 3
i p ∑
converges
' n a ∑
converges
⋅ Yes Yes
3 4
i−
⋅ ( ) 1
5 6
⋅ Yes Yes
6 7
i−
⋅ ( ) 1
10 11
⋅ Yes Yes
11 ⋅ 12 Yes Yes
3 No consecutive digits 0 0 Yes Yes
4 No consecutive digits 3 ( ) 1
1
i−
Yes Yes
1
7 No consecutive digits 64 i−
⋅ ( ) 1
4
12 No consecutive digits 1
11 9 i−
⋅ ( ) 1
9
i−
12
3 Only consecutive digits 1
23 i−
⋅ 1 No
4 Only consecutive digits 3 i ( ) 1
3
i−
4
1
7 Only consecutive digits 63 i−
1
12 Only consecutive digits 11 3 i−
4
7
i−
⋅ ( ) 1
3
7
i−
⋅ ( ) 1
3
12
i−
i−
Yes Yes
Yes Yes
Yes Yes
Yes Yes
Yes Yes
** The modifications described as “No consecutive digits” are modified series where
the denominators have no consecutive digits that are consecutive numbers. The
modifications described as “Only consecutive digits” are modified series where the
denominators have every consecutive digit being a consecutive number. Here,
numbers in base b are considered consecutive if they are adjacent to each other in the
cyclic set { 0,1,..., b −1}
. As before, every number is consecutive to itself.
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