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open problems survey
#19
TPID 17

Let f(x) be a real-entire function such that for x > 0 we have

f(x) > 0 , f ' (x) > 0 and f " (x) > 0 and also

0 < D^m f(x) < D^(m-1) f(x).

Then when we use the S9 method from fake function theory to approximate the Taylor series

fake f(x) = a_0 + a_1 x + a_2 x^2 + ...

by setting a_n x^n = f(x) ( as S9 does )

we get an approximation to the true Taylor series

f(x) = t_0 + t_1 x + ...

such that

(a_n / t_n) ^2 + (t_n / a_n)^2 = O(n ln(n+1)).

Where O is big-O notation.

reference : http://math.eretrandre.org/tetrationforu...hp?tid=863

How to prove this ?

regards

tommy1729
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Messages In This Thread
open problems survey - by bo198214 - 05/17/2008, 10:03 AM
Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM
Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM
A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM
Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM
RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM
Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM
Error terms for fake function theory TPID 17 - by tommy1729 - 03/28/2015, 10:48 PM
The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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