Taylor Series Solution to Exercise 2 - General formula:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + x f^{(1)}(0) + \frac{x^2}{2!} f^{(2)}(0) + \frac{x^3}{3!} f^{(3)}(0) + \dots$$
Given \( f(x) = \ln(1+x) \), compute derivatives at \( x = 0 \):
| n |
\( f^{(n)}(x) \) |
\( f^{(n)}(0) \) |
| 0 |
\( \ln(1+x) \) |
\( 0 \) |
| 1 |
\( (1+x)^{-1} \) |
\( 1 \) |
| 2 |
\( -(1+x)^{-2} \) |
\( -1 \) |
| 3 |
\( 2(1+x)^{-3} \) |
\( 2 \) |
| 4 |
\( -(3 \cdot 2)(1+x)^{-4} \) |
\( -(3 \cdot 2) \) |
| 5 |
\( (4 \cdot 3 \cdot 2)(1+x)^{-5} \) |
\( (4 \cdot 3 \cdot 2) \) |
Series expansion:
$$\ln(1+x) = 0 + x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$$
Compact form:
$$\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}$$
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