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1.
After 35 years as a programmer, avoiding one-off problems, I make that rookie mistake with this question: http://mathoverflow.net/questions/126739 as Douglas Zare pointed out. My first clue should have been my own statement, "This is not a sum that I expected."

A reworking of the question to start the sums at zero instead of one, would change the convergence to 1, and make everthing else confusing. So, I would like to delete the question, but some may not want their work to disappear. (I have an off-line copy.) I don't think the OP has any archival value, but what do you think?
• CommentAuthorquid
• CommentTimeApr 9th 2013

In view of the fact that some content on said post is from me perhaps I should comment, but I am not quite sure what to say.

I have certainly no objection to it being deleted. But the entire situation seems a bit surreal to me. (You cannot delete it yourself though, I think; as there are too many/too highly voted answers.)

2.
I could edit to insert the preferred version and leave the OP. The deeper meaning that underlies the series remains.
• CommentAuthorFred Kline
• CommentTimeApr 10th 2013

I have found two identies on OEIS for $m=1,2$ and I have created two that use $\gamma$, so I will update the OP in a day or two. One of the OEIS identities is stated incorrectly, so I will fix it and insert the new ones. I will also ask them about citing the OP. I have been cited there before by someone who liked something on Mathematica.SE, but I never cite my own work. Maybe I can salvage this after all.

• CommentAuthorquid
• CommentTimeApr 10th 2013

I am not sure there is a "preferred version." Starting at k=0 just feels more natural in a general sense [and makes the thing fit better with widely used things], which is I strongly believe what Douglas Zare meant with his first comment (and the second then was a reply to a not completely convincing justification).

Now, if in retrospect you think it would be more natural had you started at 0, you could say so in a comment, like: "In retrospect I agree having the series start at 0 would have been more natural. Fortunately it does not change much as regards the true underlying question as changing it always only shifts the result by 1/2. Thus I leave the question as it is."

But then your most recent comment suggest a still different development I do not fully oversee, so not sure how relevant this advice is now.

• CommentAuthorFred Kline
• CommentTimeApr 10th 2013

I just have to restrict m>0 to get k=0 to work. In looking at all the info you gave on gamma, I realized that I should do the natural method. I will also add a two line example of the deeper meaning of the k+2 factor. Like I said, I didn't expect a rational number when I ran the test.

I won't put this in the update, but what I'm trying to do is establish a connection from the Fundimental Theorem of Arithmetic to these k+2 segments. I'm not sure how to do this yet. But I'm having fun with it.

If you know of anyone who might think this one rational convergence out of thousands that contain e might be worth a paper, then tell them to go for it. My only interest is in how it might be used for Oppermann which is still undetermined. I wouldn't even want attribution because it is a serendipitous discovery.

• CommentAuthorquid
• CommentTimeApr 10th 2013

I see. (Tangentially: I should have written 1/m not 1/2 in my suggested comment.) I do not know anyone who would want to turn this into a paper.