Inf{(-1)[SUP]n[/SUP]+1/n : n=1,2,3, }=-1I was reading about

[BloodyFrozen]

inf{(-1)ⁿ+1/n : n=1,2,3,...}=-1

I was reading about Infinums and Supremums, when I saw this problem.

How did they get -1 as the Infinum?

 

[I like Serena]

Hi BloodyFrozen!

What is the lowest possible value you can think of for the expression, if you can choose any n you like?
Or perhaps more specific, what do you get if you fill in n=1, and what if n=2, and what if n=3?

 

[Studiot]

reading about Infinums and Supremums

Read again

It is infimum

As to an explanation of theinfimum or greatest lower bound:

look at the expression.

1/n is always positive. It starts at 1 for n = 1 and gets smaller and smaller as n increases.

Now consider (-1)ⁿ

Edit
When n is even (-1)ⁿ is positive and = 1 so the value of the expression for n even is positive and equals 1+ an increasingly small amount as n increases,
the largest amount we add to 1 is when n=2 thus the max value of the expression is 3/2, no matter how big we make n.

That is the supremum is 3/2 as Ilike Serena noted.

But when n is odd (-1)ⁿ is positive and = -1
So the value of the expression is -1 + an increasingly small positive value as n increases

That is the expression approaches but never reaches -1
So -1 in the infimum for the set.
Unlike with the supremum, -1 is not an member of the set, since we can get as close as we like by making n arbitrarily large.

You can write this up in posh epsilon delta and inequalities format as an exercise.

go well

 

[I like Serena]

Hi Studiot!

but it increases without limit

So there is no supremum.

Hold your horses and look again.
The supremum is 3/2.

 

[Studiot]

Thanks for saving my bacon ILS, you are quite right.
It's been a long day.

See the edit for BF's benefit.

 

[BloodyFrozen]

Thanks guys, my bad about the spelling.

So basically, we only look at (-1)ⁿ since 1/n approaches 0 at infinity?

 

[I like Serena]

Thanks guys, my bad about the spelling.

So basically, we only look at (-1)ⁿ since 1/n approaches 0 at infinity?

Basically we look at the lowest value (-1)ⁿ can take.
We look at the lowest value 1/n can take or rather approach.
And we sum those two.

 

[BloodyFrozen]

Basically we look at the lowest value (-1)ⁿ can take.
We look at the lowest value 1/n can take or rather approach.
And we sum those two.

Ok, thanks

 

[micromass]

Ok, thanks

Can you find the supremum and infimum of

• \{1/n~\vert~n>0\}
  
• \{0,2,4,6,8,10,12,...,2n,...\}
  
• \{1,0,1,0,1,0,1,0,1,0,...\}
  
• A constant sequence
  
• A sequence that monotonically increases to 1
  
• [0,1[
  

Think about this if you really want to understand what a supremum and infimum is

 

[I like Serena]

Is this Spivak-stuff?

 

[BloodyFrozen]

{1,0,1,0,1,0,1,0,1,0,...\}Is 0 infimum and 1 supremum?

[0,1] Infimum 0 Supremum 1

{0,2,4,6,8,10,12,...,2n,...} Infinum 0 and don't see how you would get the supremum

{1/n : n>0} Infimum 0 Supremum 1?

 

[I like Serena]

You seem to get the idea!

A few notes:

MM wrote [0,1[ by which he undoubtedly meant the interval that does not contain 1.
However, the supremum is still 1.

Where you don't see how to get the supremum, the supremum does not exist, or you might also say that it's infinite.

I'd consider {1/n : n>0} a trick question (I don't know if that's what MM intended), since it is not specified that n is a whole number...
Anyway, taking things literally, the supremum would be infinity.

 

[BloodyFrozen]

Thanks, how would you figure these two out?
Does a sequence have a infimum or supremum if they're the same thing?

A constant sequence

A sequence that monotonically increases to 1

Is that [0,1[ the same thing as [0,1)?

 

[I like Serena]

Does a sequence have a infimum or supremum if they're the same thing?

Sure!

A constant sequence

A sequence that monotonically increases to 1

Can you give examples of these sequences?

Is that [0,1[ the same thing as [0,1)?

No.
[0, 1] is the interval of real numbers from and including 0 up to and including 1.
[0,1[ which is also written as [0, 1) is the interval of real numbers from and including 0 up to and excluding 1.

 

[BloodyFrozen]

Sure!




Can you give examples of these sequences?




No.
[0, 1] is the interval of real numbers from and including 0 up to and including 1.
[0,1[ which is also written as [0, 1) is the interval of real numbers from and including 0 up to and excluding 1.

I should of made it clearer but I meant if [0,1[ is equal to [0,1) -> cleared up though

Constant Sequence- {1,1,1,1,1,1,1,} Infimum 1 Supremum1

Monotonically to 1- {1/8,1/7,1/6,1/5,1/4,1/3,1/2,1} Infinum 1/8 Supremum 1

I think that's what micromass was asking for?

 

[I like Serena]

I should of made it clearer but I meant if [0,1[ is equal to [0,1) -> cleared up though

Constant Sequence- {1,1,1,1,1,1,1,} Infimum 1 Supremum1

Monotonically to 1- {1/8,1/7,1/6,1/5,1/4,1/3,1/2,1} Infinum 1/8 Supremum 1

I think that's what micromass was asking for?

Yep!

To formulate it a little bit sharper:

a constant sequence has an infimum that's the same as the supremum which is the constant.

a monotonically increasing sequence to 1 has an infimum that is the first element and a supremum that is 1.

 

[BloodyFrozen]

Yep!

To formulate it a little bit sharper:

a constant sequence has an infimum that's the same as the supremum which is the constant.

a monotonically increasing sequence to 1 has an infimum that is the first element and a supremum that is 1.

Yay!

Thanks for everyone's help

 

[micromass]

Yay!

Thanks for everyone's help

Seems like you got the idea! Congratz!