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Which of the following is Cauchy's sequence ?a)(-1)^n+1/n  b)(-1)^n.n c) n+1...

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farahyasmin | Student, Undergraduate | Honors

Posted December 20, 2012 at 6:52 PM via web

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Which of the following is Cauchy's sequence ?
a)(-1)^n+1/n  b)(-1)^n.n c) n+1/n d) (-1)^n.1/n

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted December 27, 2012 at 10:06 PM (Answer #1)

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a)

-1+1, 1+1/2, -1+1/3, 1+1/4, -1+1/5, 1+1/6, -1+1/7, 1+1/8

every second number is getting close to 1, and the other numbers are getting close to -1.

The 1/n part of the expression is getting close to 0, but the (-1)^n part of the expression is bouncing between -1 and 1

Thus this sequence is not Cauchy.

Definition of Cauchy: for every epsilon, there is some N such that, if n,m>N then An and Am are within epsilon of each other.

Pick epsilon = 1

No matter what N you pick, if you look at An and A(n+1), they will be about 2 apart: one will be close to 1 and one will be close to -1, and the difference between these will be more than 1

 

b)

-1, 2, -3, 4, -5, 6, -7, 8, ...

These numbers are unbounded, so they are not convergent.  In the real numbers, a Cauchy sequence is the same as a convergent sequence.  This sequence is unbounded, so it is not Cauchy.

 

c) 1+1, 2+1/2, 3+1/3, 4+1/4, 5+1/5, ...

Again, these numbers are unbounded, so they are not Cauchy

 

d)

-1, 1/2, -1/3, 1/4, -1/5, 1/6, ...

These numbers are converging to 0.  Again, in the real numbers, Cauchy is convergent, so this is a Cauchy sequence.  Or, to see this rigorously:

Pick some `epsilon` .  We want to find N such that, for n,m>N, `|A_n - A_m|<epsilon`

But:

`|A_n - A_m| <= |A_n| + |A_m| = 1/n + 1/m < 2/m`

(if we assume m `<=` n ... one of them has to be less than or equal to the other)

Thus, we want `2/m < epsilon`

 

So: Pick N so that `N>2/epsilon`

Then, if m, n>N we have `1/m < epsilon / 2`  and `1/n<epsilon /2`

So:

`|A_n-A_m| <= 1/m+1/n < epsilon/2 + epsilon/2 = epsilon`

 

Thus, sequence 4 is Cauchy

 

 

 

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