How would I expand the function in a series of powers of x-2?
@HallsofIvy: I don’t think your answer is correct. When you have and , then your series converges to , which is not the original function.
If you want to use a geometric series method, you could use and , but then your power series is no longer in factors of (x-2).
Or you could stop at , and simply let and , which gives my original answer.
@Mattpd: My method is quite a brute force method, which one often uses when learning about power series. Basically, Taylor’s theorem states that under certain conditions, , where is shorthand for the th derivative of evaluated at . Depending on what function you have, it may be possible to find (by induction) an exact expression for .
The method HallsofIvy gave (which is correct despite a slight haste in the actual solution), reduces the problem to one for which the answer is known (namely it tries to make your function look like the expression for the sum of a geometric series).