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

  • Thread starter Thread starter AKJ1
  • Start date Start date
  • Tags Tags
    Domain Sketch
Click For Summary

Homework Help Overview

The problem involves determining the domain of the function f(x,y) = ∑(x/y)^n, where n ranges from 0 to infinity, on the xy plane. Participants are tasked with sketching the domain based on their findings.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the implications of the ratio x/y being less than or greater than 1, considering the use of geometric series for defining the function. Questions arise regarding the behavior of the function when the ratio exceeds 1 and the conditions under which the sum exists.

Discussion Status

Some participants have provided guidance on the conditions for the existence of the sum, noting that the absolute value of the ratio must be considered. There is an acknowledgment of the need to exclude combinations where the ratio is greater than 1, as well as the case where y equals 0.

Contextual Notes

Participants are exploring the implications of negative values for x and y, and the discussion includes considerations of domain violations based on the defined conditions for the function.

AKJ1
Messages
43
Reaction score
0

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.
 
Physics news on Phys.org
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.
 

Similar threads

Find f(x,y) given partial derivative and initial condition
  • · Replies 6 ·
Replies
6
Views
2K
Prove if ##x<0## and ##y<z## then ##xy>xz## (Rudin)
  • · Replies 2 ·
Replies
2
Views
2K
Domain of definition of the function f(x,y)=x^y
  • · Replies 2 ·
Replies
2
Views
1K
Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##
  • · Replies 5 ·
Replies
5
Views
1K
Is xy>0 Sufficient to Determine the Domain of f(x, y) = xy√(x²+y)?
  • · Replies 7 ·
Replies
7
Views
2K
Find Domain of f(x,y)=sqrt(5xy-2y^2)
  • · Replies 15 ·
Replies
15
Views
2K
Solve p = P(2X <= Y^2) using double integral
  • · Replies 4 ·
Replies
4
Views
2K
##G## is Injective ##\iff ## ## f ## is onto ## Y ## (Lambda Notation)
  • · Replies 3 ·
Replies
3
Views
1K
Domain of ln(x+y) | x ≠ 0, y ≠ 0 or (x,y) ≠ (0,0)?
  • · Replies 24 ·
Replies
24
Views
3K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K