# How can you find the taylor expansion of #ln(1-x)# about x=0?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the Taylor expansion of ln(1-x) about x = 0, we can use the formula for the Taylor series expansion of a function:

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

In this case, f(x) = ln(1-x) and a = 0. We need to find the derivatives of ln(1-x) at x = 0:

f(x) = ln(1-x) f'(x) = -1/(1-x) f''(x) = 1/(1-x)^2 f'''(x) = -2/(1-x)^3 f''''(x) = 6/(1-x)^4

Now, we evaluate these derivatives at x = 0:

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 f''''(0) = 6/(1-0)^4 = 6

Now, we substitute these values into the Taylor series formula:

ln(1-x) = 0 - 1*x + 1*x^2/2 - 2*x^3/3 + 6*x^4/4 + ...

Simplifying, we get:

ln(1-x) = -x - x^2/2 - x^3/3 - x^4/4 + ...

So, the Taylor expansion of ln(1-x) about x = 0 is -x - x^2/2 - x^3/3 - x^4/4 + ...

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the power series of #f(x)= ln(5-x)^2#? What is its radius of convergence?
- How do you find the power series representation for the function #f(x)=e^(x^2)# ?
- How do you find the radius of convergence #Sigma x^(3n)/5^n# from #n=[1,oo)#?
- What is the interval of convergence of #sum_1^oo xsin((pi*n)/2)/n #?
- How do you find the Maclaurin Series for # f(x)= x sinx#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7