Understanding the Supremum and Infimum of 1/2n

In this article, we will explore the concepts of supremum and infimum with a specific focus on the sequence an 1/2n where n is a positive integer. This sequence is of great interest in mathematical analysis and has important applications in various fields, from calculus to computer science.

Sequence Definition and Behavior

The sequence an 1/2n is defined for positive integers n. As n increases, the value of an decreases, approaching 0 but never actually reaching it. This behavior is characteristic of the sequence and will help us determine its suprema and infima.

Infimum of the Sequence

The infimum of a set is the greatest lower bound. For the sequence an 1/2n, as n increases, the values of an get smaller and smaller, approaching 0 but never actually reaching it. Therefore, the infimum is the greatest value that is still a lower bound for the set of all an values. Mathematically, we can write:

tinf 1/2n 0

This means that 0 is the largest number that is still less than or equal to all values of an in the sequence.

The supremum of a set is the least upper bound. For the sequence an 1/2n, the largest value in the sequence occurs when n is smallest, which is n 1. At this point, the value is a1 1/2. Since 1/2 is greater than all other terms in the sequence, it is the least upper bound. Mathematically, we can write:

tsup 1/2n 1/2

This means that 1/2 is the smallest number that is still greater than or equal to all values of an in the sequence.

Summary of Supremum and Infimum

By examining the sequence an 1/2n, we have determined the following:

tInfimum: 0 tSupremum: 1/2

These values provide important insights into the behavior and limit points of the sequence as n increases.

Special Cases: Different Starting Indices

It is also worth considering the sequence when the index n starts at 0. In this case, the sequence would be defined differently:

tWhen n starts at 0: The supremum of the sequence is 1 because a0 1. The infimum remains the same at 0. tWhen n starts at 1: The supremum remains at 1/2 as determined earlier, but the infimum is still 0.

These special cases highlight how the choice of the starting index can impact the supremum and infimum of a sequence, while the infimum remains consistent in both cases.