What Is the Domain of f(x,y) = ∑(x/y)^n on the XY Plane?

[AKJ1]

Homework Statement

Suppose we have f(x,y) = ∑(x/y)^n , n goes from 0 to infinity. What is the domain on the xy plane? Sketch it.

3. Attempt
I was thinking to look at the scenario if x/y is less than 1, or bigger than 1. If the ratio is less than 1, then I can use an idea from geometric series to write out an explicit form for f(x,y) in which it will be defined for x>y, or in other words, below the line y = x. What about if x/y is bigger than 1? I am getting stuck here as how to represent that. Also I am not certain if the approach is even correct.

 

[ehild]

Homework Statement

Suppose we have f(x,y) = ∑(x/y)^n , n goes from 0 to infinity. What is the domain on the xy plane? Sketch it.

3. Attempt
I was thinking to look at the scenario if x/y is less than 1, or bigger than 1. If the ratio is less than 1, then I can use an idea from geometric series to write out an explicit form for f(x,y) in which it will be defined for x>y, or in other words, below the line y = x. What about if x/y is bigger than 1? I am getting stuck here as how to represent that. Also I am not certain if the approach is even correct.

The approach is almost correct. Both x and y can be negative, and you can use the sum of geometric series if |x/y|<1. Does the sum exist in the opposite case? (Is it finite?)

 

[AKJ1]

The approach is almost correct. Both x and y can be negative, and you can use the sum of geometric series if |x/y|<1. Does the sum exist in the opposite case? (Is it finite?)

Oh youre right! I keep forgetting we look at the absolute value of the ratio.

The sum does not exist in the opposite case, therefore any such combination where the ratio is greater than 1 is a domain violation and should be excluded from the sketch?

 

[ehild]

The sum does not exist in the opposite case, therefore any such combination where the ratio is greater than 1 is a domain violation and should be excluded from the sketch?

Yes.And do not forget y=0, when the ratio does not exist.