Friday, 7 August 2015

sequences and series - Does sumlimitsinftyn=1frac1Pnln(Pn) converge to the golden ratio?



The sum n=21nln(n) does not converge.



But the sum n=11Pnln(Pn) where Pn denotes the nth prime number appears to be.



Is that correct, and if so, how can we calculate the value of convergence?



Is it possible that this sum converges to the golden ratio (1+52)?


Answer




With Pnnln(n), we should have N1Pnln(Pn)Ndxxln(x)2=1lnN
If the sum for n up to π(19999999)=1270607 is 1.57713, we'd expect
the remainder to be about .071, which would push the total to about 1.648, too high for ϕ.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...