Question 14 in 12.10 asks:
Find the Taylor series for f(x) centered at a=2 where f(x)=lnx
I started by computing the first few derivatives of the function:
f^0(x) = lnx
f^1(x) = 1/x
f^2(x) = -1/x^2
f^3(x) = 2/x^3
f^4(x) = -6/x^4
So writing this now in the form of a Taylor series:
Then f(x) = (ln2)+[(1/2)(x-2)]/1!-[(1/2^2)(x-2)^2]/2! +[(2/2^3)(x-2)^3]/3!...
Writing this as a sum:
ln2 + Sigma(1,inf) [ (-1)^(n+1) (n-1)! (x-2)^n / 2^n n! ]
Is this looking correct? I am somewhat concerned about starting the counter at n=1 and what to do with f^0(x) = lnx.
Looks fine. Since n=0 does not fit nicely within the rest of the pattern , it is good to take that term out of the sum, as you did.
By the way, if you go to http://webware.cc.umanitoba.ca:8080/webMathematica/MY.html and click on the link for the sequences and series course, you will access a bunch of interactive scripts that could be used when you want to check your answer. I have not updated these pages for a while, but they are still usable.