Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

Question

Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

f(x) = (x^2+1) / (6x-1)

The answer blanks for this problem are set up different then the others and I think it's confusing me on how to solve this problem

the format the assignment wants

____ + sum infinity n = 2 (____)

Answer 1

f(x) = (x² + 1)/(6x - 1) = x/6+ 1/36 - 37/36·1/(1 - 6x) = x/6 + 1/36 - 37/36·∑_n=0^∞(6x)ⁿ.

-1< 6x < 1; - 1/6 <x < 1/6