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