Expanding f(x,y) with Double Taylor Series

[Karlisbad]

Let be an analytic function f(x,y) so we want to take its Taylor series, my question is if we can do this:

-First we expand f(x,y) on powers of y considering x a constant so:

f(x,y)= \sum_{n=0}^{\infty}a_{n} (x)y^{n}

and then we expand a(n,x) for every n into powers of x so we have..

f(x,y)= \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b_{n}x^{m} y^{n}

I don't know if the result will be the same that taking the "Double Taylor series " for the function f(x,y)

 

[matt grime]

b_n should be b_m, and you're just asking about when you can reorder summation which is a well known property of certain sums, and not others.

 

[mathman]

b_n should be b_n,m.