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The problem you want help with is Exercise 6:

**\( T_n[\log(1+x)] = \sum_{k=1}^{n} \frac{(-1)^{k+1}x^k}{k} \)**.

This expression is the **Taylor polynomial** for the function \(\log(1 + x)\) around \(x = 0\) (Maclaurin series), and the task asks you to find the polynomial of degree \(n\).

### Step-by-step approach:

#### 1. Recognize the Taylor series expansion of \(\log(1 + x)\):
The Taylor series for \(\log(1 + x)\) at \(x = 0\) is given by:
\[
\log(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots
\]
This is an alternating series with the general term:
\[
(-1)^{k+1} \frac{x^k}{k}, \quad \text{for} \ k = 1, 2, 3, \dots
\]

#### 2. The general term:
The general form of the Taylor polynomial of degree \(n\) for \(\log(1 + x)\) is the sum of the first \(n\) terms:
\[
T_n[\log(1 + x)] = \sum_{k=1}^{n} \frac{(-1)^{k+1}x^k}{k}
\]
This matches the series you provided.

#### 3. Example with a specific \(n\):
Let's compute this for \(n = 4\):
\[
T_4[\log(1 + x)] = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}
\]
For \(n = 5\), it becomes:
\[
T_5[\log(1 + x)] = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5}
\]

#### 4. General Procedure:
To compute \(T_n[\log(1 + x)]\), simply follow the general formula and substitute the degree \(n\), computing each term up to \(x^n\).

### Summary:
- The Taylor polynomial \(T_n[\log(1 + x)]\) is given by the formula \(\sum_{k=1}^{n} \frac{(-1)^{k+1}x^k}{k}\).
- Plug in any value of \(n\) to obtain the specific Taylor polynomial of degree \(n\).

Let me know if you'd like to explore any particular part of this further!

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**Questions:**
1. How is the general term for the Taylor series derived?
2. What is the radius of convergence for the Taylor series of \(\log(1 + x)\)?
3. Can you find \(T_6[\log(1 + x)]\)?
4. How does the Taylor polynomial approximate \(\log(1 + x)\) for values of \(x\) near zero?
5. Can you visualize how the series converges as \(n\) increases?

**Tip**: The Taylor series for \(\log(1 + x)\) converges for \(-1 < x \leq 1\), so keep this in mind when using it for approximation!

Learn how to compute the Taylor series expansion for log(1 + x) up to degree n. This problem involves logarithmic functions, series expansion, and the application of the Taylor theorem.

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