The discussion focuses on the union of two sets defined by specific intervals as n approaches infinity. The first set, U [2 + 2/n, Pi - 1/n], reveals that as n increases, the intervals converge, with critical points identified at n=1 and n=2 showing empty sets. The second set, U [1/(1+n), 1/n], demonstrates a similar trend, with endpoints converging to zero as n increases. The analysis emphasizes the importance of understanding the behavior of the endpoints in relation to the union of the intervals.
PREREQUISITESMathematics students, educators, and anyone interested in advanced calculus concepts, particularly those studying set theory and limits.
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