or, more precisely:

where M is the so-called Meissel-Merten's constant, with value approximately equal to 0.2614972128476427837554268386086958590516...

This result is known as Merten's second theorem.

]]>Some standard work shows we can open the brackets

I'm not sure if I've hear the phrase "open the brackets" before. Does this mean something common and it's just the slang I'm stuck on, or is this something I haven't learned yet?

]]>Edited to add: And by "calc" I believe they were referring to analysis, as I don't think that statement would be very useful in any introductory calculus class.

]]>I have taken calc, but I don't remember that, or have never seen it. If I could get past that, the rest pretty much makes sense.

]]>I only sorta get that.

Which statement do you not get?

]]>As long as we're on the topic of infinate series, I played with the series:

1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + 1/ 13 + 1/21 +...

The reciprocals of Fibonacci numbers. I brute forced it in excel and it definately seemed to converge...

EDIT: I spoke too soon, the wikipedia article says that it does converge but no one knows how to express the answer as anything but an estimate of its value: 3.35988...

]]>http://mathworld.wolfram.com/HarmonicSe … rimes.html

confirms the wiki article but doesn't really say why.

http://www.everything2.com/index.pl?node_id=1537535

gives a proof I don't understand.

]]>EDIT: It does in fact say that, I still don't know how to prove it.

]]>I'm wondering if anyone knows (or can work out) whether it converges or diverges. And if it converges, roughly to what?

I know that the series of 1/x diverges, and that the series of 1/x² converges, and 1/P seems to be somewhere between those two.

I started comparing it with other series to see if I could get some insight, and in doing so I made up a proposal that seems true but I have no idea whether it is.

Basically, an infinite series made up of terms of 1/f(x) will always converge if you can find an integer k such that f(k+1) - f(k) is greater than any given N.

It's easy to see how this is true for x² and false for x, but for the primes it doesn't help as much.

So all in all, I've had lots of thoughts but none of them really help.

Can anyone shed light on this?