When \( x \) is small, we can use a Taylor series expansion to approximate \((1+x)\ln(1+x)\).

The Taylor series expansion of \(\ln(1+x)\) around \(x = 0\) is:

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

Multiplying this by \(1+x\), we get:

\[
(1+x)\ln(1+x) = (1+x)\left( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \right)
\]

Distributing \(1+x\) through the series:

\[
(1+x)\ln(1+x) = x + x^2 - \frac{x^3}{2} + \frac{x^4}{3} - \cdots + x \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \right)
\]

\[
= x + x^2 - \frac{x^3}{2} + \frac{x^4}{3} - \cdots + x^2 - \frac{x^3}{2} + \frac{x^4}{3} - \cdots
\]

Combining like terms:

\[
(1+x)\ln(1+x) = x + (x^2 + x^2) - \left( \frac{x^3}{2} + \frac{x^3}{2} \right) + \left( \frac{x^4}{3} + \frac{x^4}{3} \right) - \cdots
\]

\[
= x + 2x^2 - x^3 + \frac{2x^4}{3} - \cdots
\]

When \(x\) is very small, higher-order terms (\(x^3, x^4, \ldots\)) become negligible. Therefore, for small \(x\), we can approximate:

\[
(1+x)\ln(1+x) \approx x + 2x^2
\]

This approximation becomes more accurate as \(x\) approaches 0.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the Taylor series expansion for \(e^x\)?
2. How do you derive the Taylor series for \(\sin(x)\) and \(\cos(x)\)?
3. What are the applications of Taylor series in numerical analysis?
4. How do you approximate \(\ln(1-x)\) using a Taylor series?
5. How does the Maclaurin series relate to the Taylor series?
6. What is the radius of convergence for a Taylor series?
7. How do you use Taylor series to approximate definite integrals?
8. What are the limitations of using Taylor series for approximations?

**Tip:** When working with Taylor series, always consider the range of \(x\) for which the series converges to ensure the accuracy of your approximations.

Learn how to approximate (1+x)ln(1+x) for small x using a Taylor series expansion. Understand the series multiplication and the accuracy of the approximation as x approaches 0. Suitable for advanced high school students and anyone interested in mathematical approximations.

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