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To determine the Maclaurin series for \( \tan(\ln(1 + x)) \) up to the \( x^3 \) term in part (c), we can proceed as follows:

1. **Use Previous Results:**
   In parts (a) and (b), we obtained the Maclaurin series expansions for \( \cos(\ln(1 + x)) \) and \( \sin(\ln(1 + x)) \). For this question, we need the Maclaurin series for \( \tan(\ln(1 + x)) = \frac{\sin(\ln(1 + x))}{\cos(\ln(1 + x))} \).

2. **Maclaurin Series for \( \sin(\ln(1 + x)) \) and \( \cos(\ln(1 + x)) \):**
   From part (a), we have:
   \[
   \cos(\ln(1 + x)) = 1 - \frac{1}{2}x^2 + \frac{1}{2}x^3 - \frac{5}{12}x^4 + \cdots
   \]
   From part (b), we have:
   \[
   \sin(\ln(1 + x)) = x - \frac{1}{2}x^2 + \frac{1}{6}x^3 + \cdots
   \]

3. **Form the Quotient \( \frac{\sin(\ln(1 + x))}{\cos(\ln(1 + x))} \):**
   We want to expand \( \tan(\ln(1 + x)) \) using the quotient:
   \[
   \tan(\ln(1 + x)) = \frac{\sin(\ln(1 + x))}{\cos(\ln(1 + x))}.
   \]

4. **Use Series Division:**
   Perform the division of \( \sin(\ln(1 + x)) \) by \( \cos(\ln(1 + x)) \), keeping terms up to \( x^3 \):

   \[
   \tan(\ln(1 + x)) = \frac{x - \frac{1}{2}x^2 + \frac{1}{6}x^3}{1 - \frac{1}{2}x^2 + \frac{1}{2}x^3}.
   \]

   To divide, write this as:
   \[
   \tan(\ln(1 + x)) = \left(x - \frac{1}{2}x^2 + \frac{1}{6}x^3\right) \cdot \left(1 + \frac{1}{2}x^2 - \frac{1}{2}x^3\right).
   \]

5. **Multiply Out Terms up to \( x^3 \):**
   Expanding the product:
   \[
   \tan(\ln(1 + x)) = x - \frac{1}{2}x^2 + \frac{1}{6}x^3 + \frac{1}{2}x^3 - \frac{1}{4}x^4 + \cdots
   \]
   Simplify terms up to \( x^3 \):
   \[
   \tan(\ln(1 + x)) = x - \frac{1}{2}x^2 + \frac{2}{3}x^3 + \cdots
   \]

Thus, the Maclaurin series for \( \tan(\ln(1 + x)) \) up to the \( x^3 \) term is:
\[
\tan(\ln(1 + x)) = x - \frac{1}{2}x^2 + \frac{2}{3}x^3 + \cdots
\]

Would you like further details or have any questions?

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Here are some related questions to explore:

1. How is the Maclaurin series for \( \ln(1 + x) \) derived?
2. What is the general approach to finding Maclaurin series for composite functions?
3. How does term-by-term differentiation affect a Maclaurin series?
4. Can the series expansion be applied to functions other than trigonometric and logarithmic functions?
5. How do higher-order terms affect the accuracy of approximations in Maclaurin series?

**Tip:** In series division, remember to multiply each term in the numerator by each term in the denominator (up to the desired degree), simplifying the result term by term.

Learn how to determine the Maclaurin series for tan(ln(1 + x)) up to the x^3 term. This problem involves concepts of Taylor and Maclaurin series, focusing on expansions for trigonometric and logarithmic functions. Suitable for high school and introductory college calculus students.

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