Let's look again at what we did in the first part :
We use truncated taylor series to find the a_n.
since g(1) should be close to a_0 + a_1 + a_2 + ....
we can say a_0 < g(1) ( and a_0 < f(1) ofcourse )
next we set
a_0 + a_1 x + a_2 x^2 + ...+ a_n x^n < f(x) or a_0 + a_1 x + a_2 x^2 + ...+ a_n x^n < g(x)
We truncate
a_n x^n < a_0 + a_1 x + a_2 x^2 + ...+ a_n x^n < g(x)
and then we do a manoever , we set
a_n x^n < g(x)
and take it more sharply to get an upper bound :
a_n x^n = g(x)
and from there
a_n = g(x) / x^n
a_n = min [ g(x)/x^n ]
where the minimum is consider for all real x > 0.
This is already very interesting. You could do alot with that.
since both a_n and g(x)/x^n are positive for x >= 0 we can consider this
a_n = g(x) / x^n
ln(a_n) = ln(g(x)) - n ln(x)
ln(a_n) = min [ ln(g(x)) - n ln(x) ]
So let t_n(x) = ln(g(x)) - n ln(x)
Then
ln(a_n) = min [ ln(g(x)) - n ln(x) ] = min [ t_n(x) ]
Now a minimum must occur when the derivative is zero.
min [ t_n(x) ] = t_n(q_n)
where q_n is the solution to d/dx t_n(x) = 0.
d/dx t_n(x) = g'(x)/g(x) - n/x.
g'(q_n)/g(q_n) - n/q_n = 0
q_n g'(q_n)/g(q_n) = n
So q_n is a function - or inverse function - of n here.
Finally
ln(a_n) = t_n(q_n)
f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...
How could we improve that ?
Well we could use a linear rescaling.
No, not C f(x), but a better way.
f_2(x) = b_0 + b_1 x + b_2 x^2 + b_3 x^3 + ...
b_n = a_n [ f( exp( q_n ) / g( exp( q_n ) ) ]
( with b_0 < g(1) )
Now we have that
lim x -> +oo f_2(x) / g(x) = 1.
***
Keep in mind fake function theory seeks a general way.
For specific cases there might be alternative ways or better ways.
Often this involves using known functional equations.
For instance
g(x) = 1 + x + x^2/sqrt(2!) + x^3/sqrt(3!) + x^4/sqrt(4!) + ...
The method above works.
But inverting the function sqrt(n!) and similar ones, can not be done with standard functions, so the final result cannot be stated in standard functions.
So they require approximations themselves and some additional work.
An alternative for this type is by noticing sqrt(n!) is related to (n/2)!
This link here explains it in detail :
Did (https://math.stackexchange.com/users/6179/did), How fast does this grow: $f(x) =\sum\limits_{i=1}^{\infty} \dfrac{x^i}{i!^{1/2}}$ for real $x$?, URL (version: 2012-10-05): https://math.stackexchange.com/q/207721
Did's answer technique can be generalized a bit. But not to all functions.
So we wanted a better method than Did's and even better than the linear rescaling of the basic idea mentioned above.
We will describe such a method in part 3 !