liminf, limsup, limit

**TXGirl** · May 14th 2008, 03:25 PM

Hi... I am trying to understand liminf and limsup of sequences.

If I have the sequence

${\frac{1+k}{1+e^k}}$ ${+ sin(k)}<br />$

what is my liminf and my limsup?

The limit itself does not exist, so the two should necessarily be unequal, right?

I know that

$limsup=inf_{k \ge 1}$ $sup_{n \ge k}$ ${\frac{1+n}{1+e^n}}$ $+sin(n)$

but how do I determine the sup, if it exists?

Also, for the sequence

${(2+(-1)^k)^k}$ ,

am I correct that neither the limit nor the limsup exist but that ${liminf=1}$ ?

And finally, for the sequence

${1 + {\frac{(-1)^k}{k^2}}}$

because the ${lim=1}$ , that means that ${liminf=limsup=1}$ , right?

Many thanks!

**Plato** · May 14th 2008, 03:56 PM

Hi TXGirl
You need to learn to use LaTex so that we can read your questions.
I know that is is a real pain, but it is well worth it.

**TXGirl** · May 14th 2008, 04:34 PM

Originally Posted by Plato

Hi TXGirl
You need to learn to use LaTex so that we can read your questions.
I know that is is a real pain, but it is well worth it.

It's true.... I read a tutorial and changed the posting....

LaTeX Tutorial

**TXGirl** · May 15th 2008, 11:05 AM

Originally Posted by TXGirl

Hi... I am trying to understand liminf and limsup of sequences.

If I have the sequence

${\frac{1+k}{1+e^k}}$ ${+ sin(k)}<br />$

what is my liminf and my limsup?

The limit itself does not exist, so the two should necessarily be unequal, right?

I know that

$limsup=inf_{k \ge 1}$ $sup_{n \ge k}$ ${\frac{1+n}{1+e^n}}$ $+sin(n)$

but how do I determine the sup, if it exists?

Also, for the sequence

${(2+(-1)^k)^k}$ ,

am I correct that neither the limit nor the limsup exist but that ${liminf=1}$ ?

And finally, for the sequence

${1 + {\frac{(-1)^k}{k^2}}}$

because the ${lim=1}$ , that means that ${liminf=limsup=1}$ , right?

Many thanks!

Anybody have any ideas?

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