find the taylor series of a function of two variables

Hello,

I was given this practice question...

Find the Taylor Series for f(x,y)=1/(2+xy^{2}).

I though this would be easy but I can't figure out how they got the answer that they did in the back of the book.

I tried to find the pattern by finding f1 and f2, then f11, f12 and f22. Then I put it into the form of a taylor polynomial to find the pattern.

After this point I ended up with 1/2 + (3xy^{2})/4 + (5x^{2}y^{4} - 4xy^{2})/8 ...

The answer in the back of the book is: Sum n=0 ->inf (-1)^{n} x^{n}y^{2n}/2^{(n+1)}

I understand the (-1)^{n} and the 2^{(n+1)} and for the most part I can see the x^{n}y^{2n }part also but something is off somewhere.

If anyone can give me any feed back I would really appreciate it.