What is the domain of the given function when expressed in terms of x and y?

[soopo]

Homework Statement

What is the domain of the following function?

$$f(x,y) = (\sqrt(x) + x\sqrt(y)) \sum_{k=1}^{\infinity} (k^3 + k) x^k y^k$$
when $D_f \subset \Re^2$.

The Attempt at a Solution

The domain is $x>0 \in \Re$ and $y>0 \in \Re$ in my opinion.

 

[Mark44]

If f(x, y) were defined only as $\sqrt{x} + x\sqrt{y}$, that would be correct. However, I think you need to take the summation into account, and determine for which x and y the series converges.

 

[soopo]

If f(x, y) were defined only as $\sqrt{x} + x\sqrt{y}$, that would be correct. However, I think you need to take the summation into account, and determine for which x and y the series converges.

The summation is to infinity.
The sum is always positive so the range is always real.
This would suggests me that x > 0 and y > 0.

However, I am uncertain, since the solution should not be that easy.

 

[Mark44]

The sum is probably convergent only for x and y in some disk centered at the origin. Outside that disk the sum would be divergent, hence the function would be undefined. It really seems to me that if you want to find the domain for this function, you need to see which values of x and y are such that the series converges.