Taylor Series for ln(x) of Degree n at 2

[ptolema]

Homework Statement

find Taylor polynomial for ln x of degree n, at 2
(P_n,2(x))

Homework Equations

P_n,1(x)= (x-1) - (x-1)²/2 + ... + (-1)^n-1(x-1)ⁿ/n

The Attempt at a Solution

there doesn't seem to be an obvious pattern to this. the coefficients for n=1 to n=4 are 2, -8, 24, -64. there is a common factor of 2, yes, but this doesn't account for the sign changes. the first term is ln 2, a positive number, then the next is (x-2)/2, another positive coefficient. how can i find the formula expressed with n and x for each term?

 

[vela]

You should have in your notes or book a general formula for a Taylor series. Start by looking that up.

 

[ptolema]

yes, i did check all my notes, but the only ones given were formulas for P_n,1(x). I'm not sure how to change it to P_n,2(x) for ln, that's what i need help with

 

[Dick]

yes, i did check all my notes, but the only ones given were formulas for P_n,1(x). I'm not sure how to change it to P_n,2(x) for ln, that's what i need help with

How did you get that the coefficients for n=1 to n=4 are 2, -8, 24, -64? That's certainly correct if you mean 1/2, -1/8, etc. Don't you see how to get the pattern for a general coefficient from that? Probably easier than deriving it from Pn,1(x).

 

[ptolema]

i see now! after staring at it for a bit, i finally realized the (rather obvious) trend, thanks