finite alternating harmonic series

• Jan 17th 2013, 08:34 PM
johng
finite alternating harmonic series
This is similar to an existing thread, but sufficiently different, I think.
For any positive integer n, let $S(n)=\Sigma^n_{k=1}\left({1\over2k-1}-{1\over2k}\right)={u_n\over d_n}$ with $u_n$ and $d_n$ relatively prime. The question concerns $u_n$ and $d_n$.

One result. The exact power of 2 that divides $d_n$ is floor(log2(n))+1.

Here's the proof:
Attachment 26598

What else can one say about the sequence $d_n$? The exact power of 3 that divides $d_n$ seems to be an increasing sequence, but I can't find any closed formula as for 2. The exact power of 5 that divides $d_n$ is not even an increasing sequence.

The integers $u_n$ and $d_n$ get quite large. I know almost nothing about the $u_n$.
$u_2$, $u_3$, $u_5$, $u_8$ and $u_9$ are prime.

Java has a pretty sophisticated probalistic prime testing algorithm. Java says with probability greater than 1-one millionth that $u_{254}$ is prime; $u_{254}$ has 221 decimal digits.

(I inadvertenly attached a thumbnail and don't know how to delete it; ignore it.)