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

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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.
 
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AKJ1 said:

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?)
 
ehild said:
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?
 
AKJ1 said:
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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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