The question involves constructing the Taylor series expansion for a given function $f(x)$ about $x = 2$, and then using it to derive the Taylor expansions for $g(x)$ and $h(x)$. Additionally, the question requires an approximation for $\ln\left(\frac{14}{9}\right)$ using the first four terms of the Taylor series for $g(x)$.

Let’s break it down step by step:

Part 1: Taylor series expansion of $f(x)$ around $x = 2$

The function is:
$f(x) = \frac{2}{1-x} + \frac{1}{1+x}.$ Expand both terms into a Taylor series around $x = 2$.

1. First term: Expand $\frac{2}{1-x}$: Using the formula for Taylor series expansion $\frac{1}{1-u} = 1 + u + u^2 + u^3 + \dots$ (valid for $|u| < 1$), write:
   $\frac{2}{1-x} = 2 \cdot \frac{1}{1-(x-2+2)} = 2 \cdot \frac{1}{1-(x-2)}.$ Expand this around $x = 2$.

2. Second term: Expand $\frac{1}{1+x}$:
   Similarly, write: $\frac{1}{1+x} = \frac{1}{1-(x-2+3)}.$ Expand this around $x = 2$.

Part 2: Construct Taylor series for $g(x)$ and $h(x)$

Using the Taylor expansion for $f(x)$, determine the Taylor series for:

• $g(x) = \ln\left(\frac{1+x}{(1-x)^2}\right)$,
• $h(x) = \frac{1 + 6x + x^2}{(1-x^2)^2}$.

Here, substitution and simplifications will be used to express $g(x)$ and $h(x)$ in Taylor-expanded form around $x = 2$.

Part 3: Approximate $\ln\left(\frac{14}{9}\right)$

Use the first four terms of the Taylor series of $g(x)$ to approximate $\ln\left(\frac{14}{9}\right)$. Substitute $x = \frac{14}{9}$ into the Taylor series and calculate.

Would you like me to go through the full detailed solution for one of these steps, or provide clarifications? Let me know! Here are five follow-up questions to deepen understanding:

1. How can the general formula for Taylor series be applied systematically for $f(x)$?
2. What are the conditions for convergence of a Taylor series expansion?
3. How do we handle logarithmic functions like $\ln\left(\frac{1+x}{(1-x)^2}\right)$ for Taylor expansions?
4. Why is substituting $x = \frac{14}{9}$ in the Taylor series of $g(x)$ valid for approximations?
5. What are practical uses of Taylor series in approximations for transcendental functions?

Tip: Always check the radius of convergence when using Taylor series approximations, especially near boundaries of convergence.