Divergence of the sum of the reciprocals of the primes

[Karamata]

Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges

i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff the sum a1+...+ak+... converges.

Sorry for bad english

 

[DonAntonio]

Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges

i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff the sum a1+...+ak+... converges.

Sorry for bad english

They are using...that theorem. If you want to see why is true take into consideration everything's positive

here and apply logarithms. Of course, you could also google "infinite products"...:>)

DonAntonio

 

[Karamata]

Well, I can say:

Let [itex]a_n>0, ~n\in \bf{N}[/itex]. We can say [itex]\displaystyle\prod_{n=1}^{+\infty}(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}\log(1+a_n)[/itex]. Also, [itex]\displaystyle\sum_{n=0}^{+\infty}\log(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}a_n[/itex], because [itex]\displaystyle\lim_{n\to +\infty}\frac{\log(1+a_n)}{a_n} = 1[/itex].

So, we have: [itex]\displaystyle\prod_{n=1}^{+\infty}(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}a_n[/itex].

BUT, here, I have this minus...?!?

 

[DonAntonio]

Well, I can say:

Let [itex]a_n>0, ~n\in \bf{N}[/itex]. We can say [itex]\displaystyle\prod_{n=1}^{+\infty}(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}\log(1+a_n)[/itex]. Also, [itex]\displaystyle\sum_{n=0}^{+\infty}\log(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}a_n[/itex], because [itex]\displaystyle\lim_{n\to +\infty}\frac{\log(1+a_n)}{a_n} = 1[/itex].

So, we have: [itex]\displaystyle\prod_{n=1}^{+\infty}(1+a_n)[/itex] converge iff converge [itex]\displaystyle\sum_{n=0}^{+\infty}a_n[/itex].

BUT, here, I have this minus...?!?

Don't worry about that: check theorem 1 here http://tinyurl.com/7cu4k4p , which sends you to check the nice book by

Knopp "Infinite sequences and series" (see chapter 3, section 7 there)

DonAntonio

 

[Karamata]

OK, it seems well.

Thanks.