Taylor's series expansion of (ln(x/a))^2

Question

I am looking for a series expansion of the function (ln(x/a))^2. (where x varies between -0.7 to 0.7) I have tried the following:

However, after inserting values for my experiment, the approximation failed woefully. I would be grateful to anyone to show me where I am getting it wrong. Thanks in advance!

Answer 1

Taylor's theorem: Let k ≥ 1 be an integer and let the function f : R → R be k times differentiable at the point a ∈ R. Then there exists a function err : R → R such that

$$[ \ f(x)= f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^2}{2!}+...+f^{(k)}(a)\frac{(x-a)^k}{k!}+err(x)(x-a)^k\]$$

with (where x varies between -0.7 to 0.7) the hypotheses of the theorem are not satisfied.