freudchicken06
New member
- Joined
- Dec 12, 2006
- Messages
- 3
I'm reviewing for my Calculus II final exam on Friday and I'd like to go over two problems I got wrong on a previous exam so that I'll know how to do them for Friday. I apologize in advance for it being a little confusing to read as I am new to this site and I'm not sure how to put in codes or anything like that to get superscripts, subscripts, summation notation, etc. ANY help you could give me on these two would be greatly appreciated!:
1.) Find the Taylor series of f(x) = ln(x) around a = 2. Give your answer in summation notation. Here's my work from the exam. Please tell me what I'm doing wrong because I really don't know:
f(x) = ln (x)
f'(x) = 1/x
f''(x) = -1/x^2
f'''(x) = -1*1*-2/x^3
f^n(x) = [((-1)^n-1)(n-1)!]/x^n
f(2) = ln(2)
f'(2) = 1/2
f''(2) = -1/4
f'''(2) = 1/4
infinity
(sigma) [(f^n(2))(x-2)^n]/n!
n = 0
infinity
(sigma) [((-1)^n-1)(n-1)!(x-2)^n]/(2^n)n!
n = 0
Then at the end, I factored out an n! from the (n-1)! in the numerator and cancelled the n! from both the numerator and the denominator leaving
infinity
(sigma) [((-1)^n-1)(n-1)(x-2)^n]/2^n
n = 0
2.) Use Taylor's Inequality to estimate ln(1.1) to within an error of 0.001. Write your answer as a single number. Also, I was given the following information:
ln(1+x) = (sigma from n=1 to infinity) ((-1)^(n+1))((x^n)/n)
Here's my work from the exam, although I hardly got past the first step. ("Estimation" problems of this form seem to give me a lot of trouble so PLEASE show as much detail on how to do this problem as you can.)
|Rn(1.1)| is less than or equal to [(f^(n+1)(a))(1.1)^(n+1)]/(n+1)! is less than or equal to 1/1000
I got to this point of setting up the inequality and I have a ton of trouble getting past here and finding a value for M. I just can't figure it out, no matter how many times I try. PLEASE help!
1.) Find the Taylor series of f(x) = ln(x) around a = 2. Give your answer in summation notation. Here's my work from the exam. Please tell me what I'm doing wrong because I really don't know:
f(x) = ln (x)
f'(x) = 1/x
f''(x) = -1/x^2
f'''(x) = -1*1*-2/x^3
f^n(x) = [((-1)^n-1)(n-1)!]/x^n
f(2) = ln(2)
f'(2) = 1/2
f''(2) = -1/4
f'''(2) = 1/4
infinity
(sigma) [(f^n(2))(x-2)^n]/n!
n = 0
infinity
(sigma) [((-1)^n-1)(n-1)!(x-2)^n]/(2^n)n!
n = 0
Then at the end, I factored out an n! from the (n-1)! in the numerator and cancelled the n! from both the numerator and the denominator leaving
infinity
(sigma) [((-1)^n-1)(n-1)(x-2)^n]/2^n
n = 0
2.) Use Taylor's Inequality to estimate ln(1.1) to within an error of 0.001. Write your answer as a single number. Also, I was given the following information:
ln(1+x) = (sigma from n=1 to infinity) ((-1)^(n+1))((x^n)/n)
Here's my work from the exam, although I hardly got past the first step. ("Estimation" problems of this form seem to give me a lot of trouble so PLEASE show as much detail on how to do this problem as you can.)
|Rn(1.1)| is less than or equal to [(f^(n+1)(a))(1.1)^(n+1)]/(n+1)! is less than or equal to 1/1000
I got to this point of setting up the inequality and I have a ton of trouble getting past here and finding a value for M. I just can't figure it out, no matter how many times I try. PLEASE help!
