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We are tasked with determining the third-degree Taylor polynomial for the function \( f \) about \( x = 2 \). The function has derivatives of all orders, and we are provided with the formula for the \( n \)-th derivative of \( f \) at \( x = 2 \):

\[
f^{(n)}(2) = \frac{n!}{n \cdot 3^n}
\]

### Steps:
1. **Calculate the derivatives** at \( x = 2 \):
   - \( f(2) = 0 \) (given)
   - \( f'(2) = \frac{1!}{1 \cdot 3^1} = \frac{1}{3} \)
   - \( f''(2) = \frac{2!}{2 \cdot 3^2} = \frac{2 \cdot 1}{2 \cdot 9} = \frac{1}{9} \)
   - \( f^{(3)}(2) = \frac{3!}{3 \cdot 3^3} = \frac{6}{3 \cdot 27} = \frac{2}{27} \)

2. **Construct the third-degree Taylor polynomial**:
   The formula for the third-degree Taylor polynomial \( T_3(x) \) centered at \( x = 2 \) is:

   \[
   T_3(x) = f(2) + f'(2)(x - 2) + \frac{f''(2)}{2!}(x - 2)^2 + \frac{f^{(3)}(2)}{3!}(x - 2)^3
   \]

   Substituting the calculated values:

   \[
   T_3(x) = 0 + \frac{1}{3}(x - 2) + \frac{1}{9 \cdot 2}(x - 2)^2 + \frac{2}{27 \cdot 6}(x - 2)^3
   \]

   Simplifying the coefficients:

   \[
   T_3(x) = \frac{1}{3}(x - 2) + \frac{1}{18}(x - 2)^2 + \frac{1}{81}(x - 2)^3
   \]

3. **Match with the given options**:
   This matches option **B**.

Thus, the correct answer is **B**.

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Would you like more details or have any questions? Here are five additional related questions:

1. What is the general formula for a Taylor series expansion?
2. How do you find the radius of convergence for a Taylor series?
3. Why is it important that a function has derivatives of all orders for a Taylor series?
4. What does the remainder term in the Taylor series tell us about the accuracy of the approximation?
5. How does shifting the center of expansion (e.g., from \( x = 0 \) to \( x = 2 \)) affect the Taylor polynomial?

**Tip**: Remember that higher-order terms in a Taylor expansion provide better approximations near the center of expansion.

Let f be a function with derivatives of all orders and for which f(2) = 0. The nth derivative of f at x = 2 is given by f^{(n)}(2) = n! / (n ⋅ 3^n). What is the third-degree Taylor polynomial for f about x = 2?

Solve a problem involving a third-degree Taylor polynomial for the function f(x) about x = 2, with provided derivatives. Learn how to apply Taylor series and derivatives to construct the polynomial.

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