To find the Taylor expansion of ln(1 + x), we can use the formula for the Taylor series expansion of a function. The formula is:

f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...

In this case, our function is ln(1 + x) and we want to expand it around the point a = 0.

Step 1: Find the 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.
And so on...

Step 2: Evaluate the derivatives at a = 0
When we evaluate the derivatives at a = 0, we get:
f(0) = ln(1 + 0) = ln(1) = 0
f'(0) = 1/(1 + 0) = 1
f''(0) = -1/(1 + 0)^2 = -1
f'''(0) = 2/(1 + 0)^3 = 2
And so on...

Step 3: Write out the Taylor series expansion
Using the formula from step 1 and the values from step 2, we can write out the Taylor series expansion of ln(1 + x) as:

ln(1 + x) = 0 + 1(x - 0)/1! - 1(x - 0)^2/2! + 2(x - 0)^3/3! + ...

Simplifying this expression gives us the Taylor expansion of ln(1 + x).

Question

To find the Taylor expansion of ln(1 + x), we can use the formula for the Taylor series expansion of a function. The formula is:

f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ...

In this case, our function is ln(1 + x) and we want to expand it around the point a = 0.

Step 1: Find the 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.
And so on...

Step 2: Evaluate the derivatives at a = 0
When we evaluate the derivatives at a = 0, we get:
f(0) = ln(1 + 0) = ln(1) = 0
f'(0) = 1/(1 + 0) = 1
f''(0) = -1/(1 + 0)^2 = -1
f'''(0) = 2/(1 + 0)^3 = 2
And so on...

Step 3: Write out the Taylor series expansion
Using the formula from step 1 and the values from step 2, we can write out the Taylor series expansion of ln(1 + x) as:

ln(1 + x) = 0 + 1(x - 0)/1! - 1(x - 0)^2/2! + 2(x - 0)^3/3! + ...

Simplifying this expression gives us the Taylor expansion of ln(1 + x).

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