# 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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### NM2

Updated on November 20, 2020

• 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 almost 7 years
First, choose a proper username, second, learn LaTeX, third, show your attempts...
• NM2 almost 7 years
im trying to fix it ..
• 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 almost 7 years
• imranfat almost 7 years
Cool, now is there also a starting value for $n$? Ah, it has already been answered.
• Irregular User almost 7 years
Why does this have a limsup and liminf tag?
• 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 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 almost 7 years
Ok , but what about - $\infty$ ? Am I right about the fact that there is no minumum ?
• 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 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 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 almost 7 years
Ok , so the minimum is $-1$ ? And actually the infimum also. Am i right ?
• Mankind almost 7 years
@Noam, right :)