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
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