Showing the sequence is bounded.

**simpleas123** · May 14th 2010, 07:24 AM

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

$ 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** · May 14th 2010, 07:34 AM

What am I missing?
$\left| {\frac{{\left( { - 1} \right)^n }} {{n^3 }}} \right| = \frac{1} {{n^3 }} \leqslant 1 $

**simpleas123** · May 14th 2010, 07:41 AM

Originally Posted by Plato

What am I missing?
$\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** · May 14th 2010, 07:57 AM

Originally Posted by simpleas123

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?

**simpleas123** · May 14th 2010, 10:54 AM

Originally Posted by Plato

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!
$ | 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.** · May 14th 2010, 12:48 PM

Originally Posted by simpleas123

Ahh, its so simple!
$ | a_n | \leqslant M $
For some M.
In this case, being that an converges to 1 therefore M = 1.

Thank you very much!

$a_n$ does not converge to 1.

**mabruka** · May 14th 2010, 03:15 PM

Leave alone for a moment the statement: "If $(a_n)_n$ is convergent then it is bounded".

Boundedness is a concept that comes before convergence,

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

$|a_n|\leq M$ for all n.

Convergence has nothing to do with boundedness so far.

Thats why Plato did what he did.

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