Taylor series!!?

Question

HELP PLEASE~~
The function ln(x) has Taylor series about a centre c of (x-1) - ((x-1)^2/2) + ((x-1)^3/3) - ((x-1)^4/4) + ....
a) What is the centre c of this Taylor series?
b) Use the ratio test to find the interval of convergence of the series. Ignore the end points.

Answer 1

The Taylor series for ln(x) about c is sum{(-1)^(n+1) (x-c)^n/n, summed from n = 1 to infinity. Since you are given a power series in (x - 1), your center is 1.

You use the ratio test on abs((n+1)-st term/n-th term); the limit of this ratio (as n -> infinity) must be LESS THAN 1 for absolute convergence. The ratio is abs((n+1)(x-1)/n), and the limit is abs(x - 1). But abs(x - 1) < 1 means -1 < x - 1 < 1, or
0 < x < 2, Therefore the interval of convergence is (0,2).

Answer 2

Well, Center is at 1: (x-c), c being the center

The taylor series for Ln(x-1) is: (SUM) (-1)^(n+1)[(x-c)^n]/n
Note: sum starts at 1 and ends at infinity

Convergence: |x-1| less than or equal to 1, x cannot = 0
^ absolute values signs