Originally Posted by

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

If I have the sequence

$\displaystyle {\frac{1+k}{1+e^k}} $$\displaystyle {+ sin(k)}

$

what is my liminf and my limsup?

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

I know that

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

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

Also, for the sequence

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

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

And finally, for the sequence

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

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

Many thanks!