Posted by **Jill** on Sunday, November 26, 2006 at 6:48pm.

f(x) =ln (1-x)

a) Compute f'(x), f''(x), f'''(x). Spot the pattern and give an expression for f ^(n) (x) [the n-th derivative of f(x)]

b) Compute the MacLaurin series of f(x) (i.e. the Taylor series of f(x) around x=0)

c) Compute the radius of convergence and determine the interval of convergence of the series in b).

d) Determine the Taylor series of f'(x) around x=0. Can you do so without using b)?

e) How would you have computed part b) if you had first done part d)?

for part a) i got, please check.

f'(x) = -1/(1-x)

f''(x) = -1/(1-x)^2

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

f^(n) (x) = -((n-1)!)/(1-x)^n

for n = 1,2,3,...

This is how far i got to, please help

f^(n)(0) = -(n-1)!

so Sum_{n=0 to infinity} x^n/n! f^(n)(0)=

-Sum_{n=1 to infinity} x^n/n

Here we have used that the zero-th derivative (i.e. the function itself) is zero for x=0.

The radius of convergence, R, is given by:

R = 1/L

where L is the limit of the absolute value of a_{n+1}/a_{n} as n--> infinity.

Inour case a_{n} = 1/n and you see that L = 1.

f'(x) = -1/(1-x). You can use that the series expoansion of this function is given by the geometric series:

1/(1-x) = Sum_{n=0 to infinity} x^(n)

If you integrate this term by term you obtain the series expansion of -log(1-x).

## Answer This Question

## Related Questions

- Calculus Derivative- Taylor Series? - let f(x)= x/x-1 find f'(x) f ''(x) and a ...
- Calc 2 taylor series - use the definition of a taylor series to find the Taylor ...
- Calculus - Taylor #2 - Find the Taylor series for f(x) centered at the given ...
- Calculus - Please.... I need your help! I posted this question yesterday and no ...
- Urgent help Maths Ms sue - Use binomial series to find the Taylor series about 0...
- calculus-- need help desperately! - The Taylor series about x=5 for a certain ...
- Calculus - Hi~ Thank you for your help! I was trying to work on a problem about ...
- Maths - Use the binomial series to find the Taylor series about 0 for the ...
- Calculus - Obtain the MacLaurin series for 1/(2-x) by making an appropriate ...
- Math - Thankyou very much! I'm having trouble with this question about taylor ...

More Related Questions