Let f(x) = \displaystyle \frac{3+6x}{x} .

Compute

We see that the first term does not fit a pattern, but we also see thatf^{(k)}(1) = for k\geq 1 .

Hence we see that the Taylor series forf
centered at 1 is given by

f(x) = 9+ \displaystyle \sum_{k=1}^\infty
(x - 1)^k .

Compute

We see that the first term does not fit a pattern, but we also see that

Hence we see that the Taylor series for

You can earn partial credit on this problem.