Showing the sequence is bounded.

Oct 2009
31
2
Hey all i thought i understood this question, but after having another look i think my solution is wrong.

"Show that the sequence {an} is bounded, where an, is given by

\(\displaystyle
a_n = \frac {(-1)^n}{n^3}
\) "

I know what if i show {an} is convergent, then it is therefore bounded.

My initial idea was to show that an was convergent by the alternating series test.

But thats the problem, its not a series.
So is my method correct, or shall i be approaching this differently?

Thanks for your help! :)
 

Plato

MHF Helper
Aug 2006
22,462
8,634
What am I missing?
\(\displaystyle \left| {\frac{{\left( { - 1} \right)^n }}
{{n^3 }}} \right| = \frac{1}
{{n^3 }} \leqslant 1
\)
 
Oct 2009
31
2
What am I missing?
\(\displaystyle \left| {\frac{{\left( { - 1} \right)^n }}
{{n^3 }}} \right| = \frac{1}
{{n^3 }} \leqslant 1
\)
Why did u just take the absolute value of an?
And i dont get what your asking me?
 

Plato

MHF Helper
Aug 2006
22,462
8,634
Why did u just take the absolute value of an?
And i dont get what your asking me?
I don't get what it is that you do not understand.
Do you understand what it means to be bounded?
 
Last edited:
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Oct 2009
31
2
I don't get what it is that you do not understand.
Do you understand what it means to be bounded?
Ahh, its so simple!
\(\displaystyle
| a_n | \leqslant M
\)
For some M.
In this case, being that an converges to 1 therefore M = 1.

Thank you very much!
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
Ahh, its so simple!
\(\displaystyle
| a_n | \leqslant M
\)
For some M.
In this case, being that an converges to 1 therefore M = 1.

Thank you very much!
\(\displaystyle a_n\) does not converge to 1.
 
Jan 2010
150
29
Mexico City
Leave alone for a moment the statement: "If \(\displaystyle (a_n)_n\) is convergent then it is bounded".


Boundedness is a concept that comes before convergence,


"A sequence \(\displaystyle (a_n)_n\) is bounded if there is a number M such that

\(\displaystyle |a_n|\leq M\) for all n.


Convergence has nothing to do with boundedness so far.


Thats why Plato did what he did.
 
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