Mr. Benorin, you see, this is a local problem: the final result is evaluated at [tex]x = a[/tex]. Besides that, [tex]f[/tex] satisfies some particular conditions, which must be considered:
(a) [tex]f(x) \neq 0[/tex], over some open interval [tex]A[/tex];
(b) [tex]f[/tex] is a series of even powers;
(c) [tex]f^{(2n+1)}(a) = 0[/tex] and [tex]f^{(2n)}(a) \neq 0[/tex], [tex]n = 0, 1, 2, ...[/tex];
The final result is a function of [tex]a[/tex], and the sum symbol, [tex]\Sigma[/tex], will not appear in the final answer.
As I've pointed,
[tex]g^{(2n)}(a)=-b_n f(a)^{-3n-2}[/tex]
Now, the task would be:
Find [tex](b_n)[/tex]
Any symbolic software may show us that the first elements of this sequence are: