taylor's expansion of ln(1 + x) at x = 0

To find the Taylor expansion of ln(1 + x) at x = 0, we can use the formula for the nth derivative of ln(1 + x) and evaluate it at x = 0.

Step 1: Find the first few derivatives of ln(1 + x). The first derivative of ln(1 + x) is 1/(1 + x). The second derivative is -1/(1 + x)^2. The third derivative is 2/(1 + x)^3. The fourth derivative is -6/(1 + x)^4.

Step 2: Evaluate the derivatives at x = 0. Plugging in x = 0 into each derivative, we get: The first derivative at x = 0 is 1. The second derivative at x = 0 is -1. The third derivative at x = 0 is 2. The fourth derivative at x = 0 is -6.

Step 3: Write the Taylor expansion. The Taylor expansion of ln(1 + x) at x = 0 is given by: ln(1 + x) = ln(1) + (1/(1!))(x - 0) + (-1/(2!))(x - 0)^2 + (2/(3!))(x - 0)^3 + (-6/(4!))(x - 0)^4 + ...

Simplifying this expression, we have: ln(1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ...

This is the Taylor expansion of ln(1 + x) at x = 0.