Can someone please explain this problem to me?

Question

itz_sid · Answer

$$\sum_{n=1}^{\infty} \sqrt[n]{6}$$
As the limit goes to infinity, this get closer and closer to 0.
So then why does this converge?

solomonzelman · Answer

If the limit of the sequence is zero, then this is the only way it might converge.

solomonzelman · Answer

Otherwise you are adding non-zero terms forever.

itz_sid · Answer

The sequence converges. 
And i wont you eventually be adding 0's to it since the limit of the series goes to 0?

solomonzelman · Answer

$\displaystyle \sum_{n=1}^{\infty} \sqrt[n]{6}$

$\displaystyle \lim_{n\to \infty}\sqrt[n]{6}=1$

solomonzelman · Answer

this limit is not zero. and here is the reason....

solomonzelman · Answer

$\displaystyle \lim_{n\to \infty}\sqrt[n]{6}=\lim_{n\to \infty}6^{1/n}$

so, as $n\to\infty$,
$1/n~~\to~~0$,
and the whole limit approaches 1.

itz_sid · Answer

Ooooooh. Wow. okay I see.

solomonzelman · Answer

More formally,

$\displaystyle \lim_{n\to \infty}\sqrt[n]{6}=\lim_{n\to \infty}6^{1/n}$

Substitution:
$\displaystyle n=1/x$    then 1/n=x   (right?)
If $\displaystyle n=1/x$ and $\displaystyle n\to \infty$, then $\displaystyle x\to 0$

so your new (equivalent) limit is:
$\displaystyle \lim_{x\to0}6^{x}=1$

solomonzelman · Answer

Yes, and then you know that since your sequence does NOT converge to zero, the series .... (blah blah blah  that you already know)

itz_sid · Answer

Yea, makes sense! :D

solomonzelman · Answer

very nice!