Modifications of the Harmonic Series - Department of Mathematics

∞
THEOREM 3: The harmonic series, ∑ n diverges in p
.
1 1/
n=
PROOF: We look at two different types of terms. First, the terms that can be
k
written as 1 / mp where m∈{0,1,..., p−1}
. These terms have p -adic norm equal to
k
p . So as n →∞ the p -adic norm of these terms goes to infinity. The rest of the
terms are not divisible by any power of p , so their p -adic norm is equal to 1. Then
lim →∞ a ≠ 0 , and hence the harmonic series diverges in p
.
n n p
COROLLARY: No sub-series of the harmonic series converges in p
.
PROOF: From the p -adic norms of the terms 1/
subsequence of an will have the property lim →∞ a = 0 .
4.2 The p-adic Harmonic Sequence
n n p
a = n,
it is clear that no
In the previous section, we saw that the harmonic series diverges in p
, and
has no converging sub-series. Here, we will look instead at the ‘harmonic’ sequence.
That is, the sequence 1, , , ... in p
.
1 1 1
2 3 4
THEOREM 4: The harmonic sequence diverges in p
.
PROOF: For a sequence a to converge in p , we need lim a − a = 0 .
We examine the subsequence
a
n
1
= . Then,
p
n n
23
n
n→∞ n+ 1 n p