Cauchy Boundedness: Partial Sums Unbounded?

[pivoxa15]

Homework Statement

Thm: If a sequence is Cauchy than that sequence is bounded.

However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded.

Sequence of partial sums is Cauchy b/c d(1/x,1/(x+1))=1/(k(k+1)) -> 0 as k->infinity

Have I done something wrong?

 

[matt grime]

Why do you think the sequences of partial sums are cauchy? They aren't.

I cannot begin to decipher what b/c d(... means.

If S_n means the sum to n terms, what makes you think that given e>0, there is an n(e) such that for all n,m>n(e) we have |S_n - S_m| < e?

It is easy to show this is impossible, using the same idea as to show that the series itself diverges. Remember that

1+1/2+1/3+1/4+1/5+... > 1+(1/2+1/2)+(1/4+1/4+1/4+1/4)+(1/8+1/8+1/8+...) > 1+1+1+1+...

 

[siddharth]

Is b/c d(.., "because" , the "distance" ...?

As matt suggested, I think you should be less cryptic in your notation.

 

[AKG]

Homework Statement

Thm: If a sequence is Cauchy than that sequence is bounded.

However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded.

Sequence of partial sums is Cauchy b/c d(1/x,1/(x+1))=1/(k(k+1)) -> 0 as k->infinity

Have I done something wrong?

Yes, that's not the definition of Cauchy. A sequence a(n) is Cauchy if for all E > 0, there exist N natural such that for all m and n > N, |a(n) - a(m)| < E. All you've shown is that for all E > 0, there exists some natural N such that |a(N) - a(N+1)| < E, or something like that.

 

[pivoxa15]

Is b/c d(.., "because" , the "distance" ...?

As matt suggested, I think you should be less cryptic in your notation.

oh... I thought it was obvious. I actually used b/c in my exam recently to indicate 'because'. Although I used it in the middle of a sentence so hopefully might be less confusing than here.

 

[matt grime]

If you'd've put a 'The' at the start of the sentence to actually make it into a sentence that might have helped. Anyway, have you understood the explanation of your mistake?

 

[pivoxa15]

I think I have. Each term in the sequence is a series summed over a finite number of terms. However the larger the number of terms, the larger the sequence gets. Very much unlike Cauchy seqences where a recquirement is that each term in the seqence decreases in magnitude the further out the sequence gets.

 

[matt grime]

I think I have. Each term in the sequence is a series summed over a finite number of terms. However the larger the number of terms, the larger the sequence gets.

it is perfectly possible to be strictly increasing and still bounded/cauchy

Very much unlike Cauchy seqences where a recquirement is that each term in the seqence decreases in magnitude the further out the sequence gets.

nope, there is nothing that states the terms have to decrease in magnitude. It is the difference between terms in the sequence that is important.