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The image contains Taylor series expressions for Exercises 3 and 4, which involve constructing Taylor polynomials \( T_n(f(x)) \). Let's break them down and solve each one.

### Exercise 3:
The given expression is:

\[
T_n(a^x) = \sum_{k=0}^{n} \frac{(\log a)^k}{k!} x^k
\]

#### **Explanation:**
- This is the Taylor series expansion for \( a^x \) around \( x = 0 \) (Maclaurin series).
- \( a^x \) can be written as \( e^{x \log a} \), and the series expansion of \( e^x \) is:

\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}
\]

- Substituting \( x \log a \) for \( x \) in the expansion of \( e^x \):

\[
e^{x \log a} = \sum_{k=0}^{n} \frac{(\log a)^k}{k!} x^k
\]

Thus, the series for \( T_n(a^x) \) is:

\[
T_n(a^x) = \sum_{k=0}^{n} \frac{(\log a)^k}{k!} x^k
\]

This is the answer for Exercise 3.

### Exercise 4:
The given expression is:

\[
T_n \left( \frac{1}{1+x} \right) = \sum_{k=0}^{n} (-1)^k x^k
\]

#### **Explanation:**
- The function \( \frac{1}{1+x} \) is the classic geometric series expansion.
- The Taylor series for \( \frac{1}{1+x} \) around \( x = 0 \) is:

\[
\frac{1}{1+x} = \sum_{k=0}^{\infty} (-1)^k x^k
\]

This is valid for \( |x| < 1 \).

- Since we are only considering the first \( n \) terms, the Taylor polynomial is:

\[
T_n \left( \frac{1}{1+x} \right) = \sum_{k=0}^{n} (-1)^k x^k
\]

This is the answer for Exercise 4.

### Final Answers:

1. For Exercise 3:
   \[
   T_n(a^x) = \sum_{k=0}^{n} \frac{(\log a)^k}{k!} x^k
   \]

2. For Exercise 4:
   \[
   T_n \left( \frac{1}{1+x} \right) = \sum_{k=0}^{n} (-1)^k x^k
   \]

Let me know if you need more details or explanations for these problems!

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Here are 5 related questions for practice:
1. How would you find the Taylor series for \( \sin(x) \)?
2. What is the general form of the Taylor series for \( e^x \)?
3. How would the Taylor series change if you expanded \( a^x \) around a point other than 0?
4. What is the radius of convergence for the series \( \sum_{k=0}^{\infty} (-1)^k x^k \)?
5. How would you find the remainder term in the Taylor series approximation for \( \frac{1}{1+x} \)?

**Tip:** When finding Taylor series, always check the radius of convergence to ensure that the series is valid in the given domain.

This article solves two Taylor series problems: one for the function a^x using the logarithmic expansion and the other for the geometric series of 1/(1+x). Detailed step-by-step explanations are provided for each series expansion.

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