08/16/2010, 11:05 PM
i almost forgot about this , but when considering paris constant i think i stumbled upon a proof.
the assumed proof shows divergence ( as i first suspected ).
first i show that it cannot converge at an exponential rate.
this is done by noting the koenigs analogue :
note that lim k -> oo eta^^k ^ z is in the neigbourhood of eta^^(k-1) and eta^^(k+1) if z is in the neighbourhood of eta.
hence lim k -> oo eta^^k ^ z ~ eta ^^k
koenigs then becomes
lim k-> oo (eta^^k - e) / Q^k
where Q is the derivate of eta^x at x = e ( since e is fixpoint ).
D eta^x = eta^x / e => eta^e / e = e/e = 1.
thus we arrive at Q = 1 and
lim k-> oo (eta^^k - e) / 1
hence eta^^k - e shrinks slower than exponential.
( for bases > eta , Q > 1 and for bases < eta , Q < 1 hence this can be used to prove the div or conv for other bases easily. )
since eta^^k - e shrinks slowly we can approximate it well with polynomials and thus we construct a taylor series.
r_k = -(eta^^k - e) = e - eta^^k
a strictly negative term sum converges iff its corresponding positive term sum converges and vice versa where corresponding means all terms multiplied by -1.
r_k = e - eta^^k
r_k+1 = e - eta^^k+1 = e - eta^eta^^k
r_k+1 = e - eta^(-r_k + e) ( because eta^^k = -r_k +e )
r_k+1 = e - eta^(-r_k + e) = e - e eta^-r_k ( because eta^e = e )
r_k+1 = e (1 - eta^-r_k)
and this is a pretty selfref about how fast r_k grows to e , now use taylor :
r_k+1 = e ( r_k/e - r_k^2 /(2e^2) + r_k^3/(6e^3) + O(r_k^4) )
now §r_k+1§ - r_k = r_k^2 /(2e) + r_k^3/(6e^2) + O(r_k^4)
compare
§e/(r+1)§ - e/r = e^2/(2e r^2) => 1/(r+1) - 1/r = - r^-2 / 2
=> 1/r - 1/(r+1) = r^-2 / 2.
which implies that since eta^^k - e shrinks slower than exponential , r_k seems close to 2e/k since by the above we can remove the /2 part by doubling our estimate of e/r :
2/r - 2/(r+1) = r^-2 2/2 = r^-2.
and of course 1/r - 1/(r+1) = r^2 approximately for 1/r hence r_k seems between 2e/(sqrt(r^2 + 3r)) + C/r^3 and 2e/(sqrt(r^2 -3r)) + C/r^3.
thus r_1 + r_2 + r_3 + ... diverges within the order of O(log(x) + C/x^2).
Q.E.D.
tommy1729
the assumed proof shows divergence ( as i first suspected ).
first i show that it cannot converge at an exponential rate.
this is done by noting the koenigs analogue :
note that lim k -> oo eta^^k ^ z is in the neigbourhood of eta^^(k-1) and eta^^(k+1) if z is in the neighbourhood of eta.
hence lim k -> oo eta^^k ^ z ~ eta ^^k
koenigs then becomes
lim k-> oo (eta^^k - e) / Q^k
where Q is the derivate of eta^x at x = e ( since e is fixpoint ).
D eta^x = eta^x / e => eta^e / e = e/e = 1.
thus we arrive at Q = 1 and
lim k-> oo (eta^^k - e) / 1
hence eta^^k - e shrinks slower than exponential.
( for bases > eta , Q > 1 and for bases < eta , Q < 1 hence this can be used to prove the div or conv for other bases easily. )
since eta^^k - e shrinks slowly we can approximate it well with polynomials and thus we construct a taylor series.
r_k = -(eta^^k - e) = e - eta^^k
a strictly negative term sum converges iff its corresponding positive term sum converges and vice versa where corresponding means all terms multiplied by -1.
r_k = e - eta^^k
r_k+1 = e - eta^^k+1 = e - eta^eta^^k
r_k+1 = e - eta^(-r_k + e) ( because eta^^k = -r_k +e )
r_k+1 = e - eta^(-r_k + e) = e - e eta^-r_k ( because eta^e = e )
r_k+1 = e (1 - eta^-r_k)
and this is a pretty selfref about how fast r_k grows to e , now use taylor :
r_k+1 = e ( r_k/e - r_k^2 /(2e^2) + r_k^3/(6e^3) + O(r_k^4) )
now §r_k+1§ - r_k = r_k^2 /(2e) + r_k^3/(6e^2) + O(r_k^4)
compare
§e/(r+1)§ - e/r = e^2/(2e r^2) => 1/(r+1) - 1/r = - r^-2 / 2
=> 1/r - 1/(r+1) = r^-2 / 2.
which implies that since eta^^k - e shrinks slower than exponential , r_k seems close to 2e/k since by the above we can remove the /2 part by doubling our estimate of e/r :
2/r - 2/(r+1) = r^-2 2/2 = r^-2.
and of course 1/r - 1/(r+1) = r^2 approximately for 1/r hence r_k seems between 2e/(sqrt(r^2 + 3r)) + C/r^3 and 2e/(sqrt(r^2 -3r)) + C/r^3.
thus r_1 + r_2 + r_3 + ... diverges within the order of O(log(x) + C/x^2).
Q.E.D.
tommy1729