The discussion focuses on the union of two sets defined by intervals as n varies over the natural numbers. Participants analyze the behavior of these intervals as n approaches infinity, exploring the limits and characteristics of the unions involved.
Participants generally agree on the need to analyze the limits and endpoints of the intervals, but there is no consensus on the final characteristics of the unions or the implications of the empty intervals.
Some assumptions about the behavior of the intervals as n approaches infinity are not fully resolved, and the discussion does not clarify the implications of the empty intervals for the unions.
Write down some of the intervals: when n= 1, [4, pi-1] is empty because pi-1< 4! when n= 2, [3,pi-1/2] is empty because pi- 1/2< 3! when n= 4, [4/3, pi- 1/3] is non-empty and is precisely that interval. It might be a good idea to draw a few of those on a number line. What is crucially important, as Pseudo Statistic said, is to see what 2+2/n and pi- 1/n converge to as n goes to infinity.ShengyaoLiang said:find:
a)
U [2 + 2/n , Pi - 1/n ]
n∈N.
"U" means union. "pi" means 3.1415926535...
When n= 1 this is [1/2,1], when n= 2, [1/3, 1/2], when n= 3, [1/4, 1/3]. Again, what happens to the endpoints as n goes to infinity? Remember that you are taking a union here.b)
U [1 / （1+n） , 1/n ]
n∈N.
thank you so much