E as an average

  • #1
3,077
4
How can one generate the sequence of whole numbers x and y which converge upon the equality

(2x+3y)/(x+y) = e

?
 

Answers and Replies

  • #2
AKG
Science Advisor
Homework Helper
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(2x+3y)/(x+y) = e gives x/y = (e-3)/(2-e), so it suffices to find a sequence of rationals which converges to (e-3)/(2-e). If the decimal expansion of (e-3)/(2-e) is given, then we can just set y = 1, 10, 100, 1000, 10000, ... and make the obvious choice for x.
 
  • #3
529
1
(2x+3y)/(x+y) = e gives x/y = (e-3)/(2-e), so it suffices to find a sequence of rationals which converges to (e-3)/(2-e). If the decimal expansion of (e-3)/(2-e) is given, then we can just set y = 1, 10, 100, 1000, 10000, ... and make the obvious choice for x.
Same as above- except use continued fractions for x/y=(e-3)/(2-e)
 

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