This task involves finding the Taylor series expansion of the function

$f(x) = \frac{1 + x + x^2}{1 - x + x^2}$

around $x_0 = 0$ up to $x^4$, using Peano's form for the remainder. Additionally, you are asked to find the coefficient of the fourth derivative, $f^{(4)}(0)$.

Steps to solve:

1. Expand numerator and denominator separately: Write Taylor expansions for $1+x+x^2$ and $1-x+x^2$.
2. Divide series (using long division or binomial expansion): Perform division of the two Taylor expansions to find the terms up to $x^4$.
3. Differentiate or analyze coefficients: Use the expanded series to find $f^{(4)}(0)$.

Let’s compute these.

Step 1: Expand numerator and denominator

The numerator $N(x) = 1 + x + x^2$ is already a polynomial, so no expansion is required.

The denominator $D(x) = 1 - x + x^2$ can be expanded using a Taylor series if needed (though it’s already in polynomial form). For simplicity, rewrite it as: $D(x) = 1 - x + x^2.$

Step 2: Perform the division $\frac{N(x)}{D(x)}$ up to $x^4$

Using polynomial division:

1. Start with: $\frac{1 + x + x^2}{1 - x + x^2}.$

2. First term: Divide $1$ by $1$ to get $1$. Subtract $1(1 - x + x^2)$: $(1 + x + x^2) - (1 - x + x^2) = 2x.$

3. Second term: Divide $2x$ by $1$ to get $2x$. Subtract $2x(1 - x + x^2)$: $2x - (2x - 2x^2 + 2x^3) = 2x^2 - 2x^3.$

4. Third term: Divide $2x^2$ by $1$ to get $2x^2$. Subtract $2x^2(1 - x + x^2)$: $2x^2 - (2x^2 - 2x^3 + 2x^4) = 2x^3 - 2x^4.$

5. Fourth term: Divide $2x^3$ by $1$ to get $2x^3$. Subtract $2x^3(1 - x + x^2)$: $2x^3 - (2x^3 - 2x^4) = 2x^4.$

Thus, the Taylor series expansion is: $f(x) = 1 + 2x + 2x^2 + 2x^3 + 2x^4.$

Step 3: Compute $f^{(4)}(0)$

From the series: $f(x) = 1 + 2x + 2x^2 + 2x^3 + 2x^4 + \dots,$

the coefficient of $x^4$ is $2$. Hence: f^{(4)}(0) = 4! \cdot \text{(coefficient of x^4)}.

Since $4! = 24$ and the coefficient of $x^4$ is $2$: $f^{(4)}(0) = 24 \cdot 2 = 48.$

Final Answer:

$f^{(4)}(0) = 48.$

Would you like a detailed explanation of Taylor series or polynomial division? Here are some related questions to explore:

1. What is the Peano remainder in Taylor series, and how is it applied here?
2. How do you systematically compute higher derivatives without polynomial division?
3. Can this problem be solved using symbolic differentiation instead?
4. How do Taylor series expansions differ for rational functions?
5. What are the computational steps for finding $f^{(n)}(x_0)$ for any $n$?

Tip: Always check the radius of convergence for a Taylor series to ensure validity in a given region.