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

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Solution 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.

Solution 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.

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Updated on November 20, 2020

Comments

  • NM2
    NM2 almost 2 years

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

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

    • imranfat
      imranfat almost 7 years
      First, choose a proper username, second, learn LaTeX, third, show your attempts...
    • NM2
      NM2 almost 7 years
      im trying to fix it ..
    • imranfat
      imranfat almost 7 years
      That's great! A nice username would also be cool. Just putting some letters kind of indicates that you are only interested in an answer, but not willing to participate in this site's community...(BTW I did not downvote)
    • NM2
      NM2 almost 7 years
      You right, Sorry about this :)
    • imranfat
      imranfat almost 7 years
      Cool, now is there also a starting value for $n$? Ah, it has already been answered.
    • Irregular User
      Irregular User almost 7 years
      Why does this have a limsup and liminf tag?
  • NM2
    NM2 almost 7 years
    Thanks for the answer ! So if i got your point, There is no minimum in this set, and the infimum is $-\infty$ ?
  • Mankind
    Mankind almost 7 years
    @Noam not quite. For instance, the number $-100$ is smaller than every element in the set: $-100<-1$, $-100<1/4$, $-100<-1/9$, and so on. But $-100$ is not the infimum - there are larger numbers that are smaller than every element of the set.
  • NM2
    NM2 almost 7 years
    Ok , but what about - $\infty$ ? Am I right about the fact that there is no minumum ?
  • Mankind
    Mankind almost 7 years
    @Noam $-\infty$ is not the infimum. The infimum is the largest number that is smaller than or equal to all numbers in the set. As I wrote above, $-100$ is an example of a number that is smaller than every element of the set. This means that your infimum is $\geq -100$. Sometimes it helps drawing in the set. Try plotting the numbers of the set on a number line, and see if you can figure out what it really looks like.
  • NM2
    NM2 almost 7 years
    Ok thanks. If i understand your point, the minimum is $\frac{-1}{n^\infty}$ ? Or this is the infinity ? im confused.
  • Mankind
    Mankind almost 7 years
    @Noam, no, the infimum is a number, like $-7$. Let me give you some other examples. The infimum of the set $\{-2,-1,0\}$ is $-2$, because $-2$ is the smallest element of this set. The set $\{-1/n\}_{n\in\mathbb{N}}$ has elements $-1,-1/2,-1/3,-1/4,...$. Notice how $-1$ is the smallest number in this set, because $-1<-1/2<-1/3<-1/4<\cdots$. This means that $-1$ is the infimum. What is the supremum then? The numbers are all less than $0$, but come closer and closer to it. Hence $0$ is the smallest number larger than all numbers in the set, and so the supremum is $0$.
  • NM2
    NM2 almost 7 years
    Ok , so the minimum is $-1$ ? And actually the infimum also. Am i right ?
  • Mankind
    Mankind almost 7 years
    @Noam, right :)