The supremum and infimum of the sequence $\{(-1)^n/n^2\}$.

Question

What is the supremum and infimum of this set ? And what are the differences between those and maximum and minimum ? Thanks.

$$\left\{\frac {(-1)^n}{n^2}:n\in\mathbb{N}\right\}$$

Answer 1

Have a look at the elements of your set $S=\{\frac{(-1)^n}{n^2}\}_{n\in\mathbb{N}}$:

$$-1,\frac{1}{4},-\frac{1}{9},\frac{1}{16},-\frac{1}{25},\ldots$$

It is just a sequence of fractions alternating in sign, with numerator $1$, and where the denominators are the squares.

The supremum is the smallest number that is at least as large as everything in this set. It looks like $\frac{1}{4}$ works. Since this is part of $S$, it is also the maximum.

The infimum is the largest number that is at least as small as everything in the set.

Answer 2

Here's a link to wikipedia for Infimum and Supremum. In short terms Infimum is the greatest lower bound and Supremum is the least upper bound. The difference from max and min is that they do not need to be elements of a set. Also sometimes a set does not have a max or a min but may have a Infimum and Supremum.