Modifications of the Harmonic Series - Department of Mathematics

series will be partitioned in the same manner as in Section 2.1 – the i th group will be
the terms that have exactly i digits in the denominator. Then we can calculate
Ni
910 i
= ⋅
−1
, and choose
M i
1
10 i−
= . It is easy to see that M i+ 1Ni+ 1 MiNi −
converges, and { MiN i}
is monotonically decreasing. So any modification that uses
this partitioning will automatically have the second conditions of both Theorem 1 and
Theorem 2 satisfied.
If we modify the series by removing the terms that contain the digit nine, then,
using the above partitioning,
ri
∑
1
89 i−
= ⋅ and therefore ( ) 1
8 9
i−
pi
9 10
= ⋅ . So, using either
Theorem 1 or Theorem 2, we see that this modification converges – as we have
already shown.
Another modification is to remove all of the terms that have two consecutive
digits alike [7]. For example, the terms 1 22and 1 75548 would be removed, but the
term 1 8384 would remain. To check for convergence we use the same partitioning
and calculate 9 and
i
also converges.
r = ( ) 1
9
i
i
p
i
−
= . Since 10
i p ∑ converges, this modified series
We can apply these theorems to many other modifications of the harmonic
r i
series. Several modifications are listed in Table 3, along with their and
i
p values
and the result of using Theorem 1 and Theorem 2 as a convergence test. The two
modifications discussed above are also included for completion. Note that all the
convergence tests for the modifications below use the same partitioning as above and
therefore have M i+ 1Ni+ 1 MiNi − ∑ converging, and { MiN i}
monotonically
decreasing.
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