# Supremum, infimum, maximum and minimum

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Hi tsr,

A=(2^n)/((2^n)+1)cos((npi)/2) such that n is an element of natural numbers. Which of the following statements is true? a)modulus of minA=modulus of maxA b)supA=infA c)modulus of supA=modulus of infA d)modulus of minA=modulus of supA

I got the max to be 1, minA to be -1, supA to be 1 and infA to be -1. So according to my answers modulus of all the above answers except b) make sense but there is only one correct answer so which one is right and how do i get to it?

Also i worked out this answer from trial and error, is there a quicker way of doing this since this was a non-calculator question.

A=(2^n)/((2^n)+1)cos((npi)/2) such that n is an element of natural numbers. Which of the following statements is true? a)modulus of minA=modulus of maxA b)supA=infA c)modulus of supA=modulus of infA d)modulus of minA=modulus of supA

I got the max to be 1, minA to be -1, supA to be 1 and infA to be -1. So according to my answers modulus of all the above answers except b) make sense but there is only one correct answer so which one is right and how do i get to it?

Also i worked out this answer from trial and error, is there a quicker way of doing this since this was a non-calculator question.

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#2

(Original post by

Hi tsr,

A=(2^n)/((2^n)+1)cos((npi)/2) such that n is an element of natural numbers. Which of the following statements is true? a)modulus of minA=modulus of maxA b)supA=infA c)modulus of supA=modulus of infA d)modulus of minA=modulus of supA

I got the max to be 1, minA to be -1, supA to be 1 and infA to be -1. So according to my answers modulus of all the above answers except b) make sense but there is only one correct answer so which one is right and how do i get to it?

Also i worked out this answer from trial and error, is there a quicker way of doing this since this was a non-calculator question.

**gn17**)Hi tsr,

A=(2^n)/((2^n)+1)cos((npi)/2) such that n is an element of natural numbers. Which of the following statements is true? a)modulus of minA=modulus of maxA b)supA=infA c)modulus of supA=modulus of infA d)modulus of minA=modulus of supA

I got the max to be 1, minA to be -1, supA to be 1 and infA to be -1. So according to my answers modulus of all the above answers except b) make sense but there is only one correct answer so which one is right and how do i get to it?

Also i worked out this answer from trial and error, is there a quicker way of doing this since this was a non-calculator question.

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#4

(Original post by

Yes

**gn17**)Yes

The inf is -1, similarly, and the minimum doesn't exist.

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(Original post by

So for n odd, ; for n even, . Clearly the sequence increases in modulus with , because the fraction term tends to 1, and 1 is an upper bound on the modulus of . Moreover, it is the supremum of the modulus. However, it is not attained for any , so the maximum does not exist.

The inf is -1, similarly, and the minimum doesn't exist.

**Smaug123**)So for n odd, ; for n even, . Clearly the sequence increases in modulus with , because the fraction term tends to 1, and 1 is an upper bound on the modulus of . Moreover, it is the supremum of the modulus. However, it is not attained for any , so the maximum does not exist.

The inf is -1, similarly, and the minimum doesn't exist.

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#6

(Original post by

But when e.g n=80, I do get A to be 1 and when n is 82 I get A to be -1.

**gn17**)But when e.g n=80, I do get A to be 1 and when n is 82 I get A to be -1.

You'll never get for any finite n.

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(Original post by

How can you possibly get A to be 1? Are you doing some sort of estimation with a calculator or something?

You'll never get for any finite n.

**davros**)How can you possibly get A to be 1? Are you doing some sort of estimation with a calculator or something?

You'll never get for any finite n.

**would not**give us 1 or -1 for any value of n, thanks for pointing that out. I know it was a non-calc question but I wasn't able to work it out without the calc, turns I out I couldn't do it with the calc as well!

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**davros**)

How can you possibly get A to be 1? Are you doing some sort of estimation with a calculator or something?

You'll never get for any finite n.

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#9

(Original post by

I just realised my mistake, yes I did use a calculator and I didn't know that a calculator would round off the answer, I thought even the calculator

**gn17**)I just realised my mistake, yes I did use a calculator and I didn't know that a calculator would round off the answer, I thought even the calculator

**would not**give us 1 or -1 for any value of n, thanks for pointing that out. I know it was a non-calc question but I wasn't able to work it out without the calc, turns I out I couldn't do it with the calc as well!
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(Original post by

It returns an answer correct to 16dp or whatever. Once the value of the expression is greater than 0.9999999999999999, it will return 1.000000000000000.

**Smaug123**)It returns an answer correct to 16dp or whatever. Once the value of the expression is greater than 0.9999999999999999, it will return 1.000000000000000.

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#11

(Original post by

Can I also ask how you know that it will never give us 1? Is it because if we take the limit as n tends to infinity we get 1?

**gn17**)Can I also ask how you know that it will never give us 1? Is it because if we take the limit as n tends to infinity we get 1?

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#12

**gn17**)

I just realised my mistake, yes I did use a calculator and I didn't know that a calculator would round off the answer, I thought even the calculator

**would not**give us 1 or -1 for any value of n, thanks for pointing that out. I know it was a non-calc question but I wasn't able to work it out without the calc, turns I out I couldn't do it with the calc as well!

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#13

(Original post by

let me give one simple advice: take your calculator home and forget about it (assuming you are studying maths at any serious uni)

**astroturf**)let me give one simple advice: take your calculator home and forget about it (assuming you are studying maths at any serious uni)

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#14

(Original post by

Well said - I haven't used a calculator in all my one-and-three-quarter years.

**Smaug123**)Well said - I haven't used a calculator in all my one-and-three-quarter years.

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#15

(Original post by

I can't even think of an undergrad topic where a calculator would be any use...

**Noble.**)I can't even think of an undergrad topic where a calculator would be any use...

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#16

(Original post by

There was one question in a numerical analysis example sheet which asked you to simulate a five-decimal-place computer. By hand. Performing approximately twenty additions, ten multiplications and five divisions.

**Smaug123**)There was one question in a numerical analysis example sheet which asked you to simulate a five-decimal-place computer. By hand. Performing approximately twenty additions, ten multiplications and five divisions.

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#18

**Noble.**)

I can't even think of an undergrad topic where a calculator would be any use...

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#20

**gn17**)

Can I also ask how you know that it will never give us 1? Is it because if we take the limit as n tends to infinity we get 1?

Whilst that is a reason why for all , it's far from the most obvious reason. The obvious reason is that a fraction is equal to one if and only if the numerator and denominator are equal, and it should be obvious that .

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