i need to prove that if f,g are analytic in (-a,a) and g(x) isnt zero for all x in (-a,a) then f/g is also analytic in some interval (-b,b). my defintiion for analytic function is that, f is analytic if there exists r>0 such that it equlas its power series in (-r,r). now in order to prove this i thought to set the next things: first, to prove that 1/g is analytic in (-1/a,1/a) and then bacuase either a>=1/a or vice versa we have by another statement that if f,g are analytic in some interval then f*g is analytic in the same interval. my problem is, if this is correct then i need a formula for n-th derivative of 1/g, in order to show that the limit of the lagrange's residue of the taylor series approaches 0 in this interval? my question what is this formula? thanks in advance.