Does the Series $\sum (1 - \frac{2}{n})^3n$ Converge or Diverge?

[nuclearrape66]

does the series from 1 to infinity of $$\sum$$(1 - (2/n))^3n convegre or diverge.

i've tried almost all the tests and can't figure it out...

 

[rocomath]

FAQ: Why hasn't anybody answered my question? - https://www.physicsforums.com/showthread.php?t=94383

 

[quasar987]

Have you even tried the simplest test of them all? Does the argument of the series goes to 0 as n increases without bounds?

 

[nuclearrape66]

i've tried the root test. does not work. meaning it's inconclusive.

and i don't understand your question.
"Does the argument of the series goes to 0 as n increases without bounds?"
are you saying do the terms get closer to zero? yes.

 

[nuclearrape66]

if you can hint me to what test i should use maybe i can figure it out.

 

[quasar987]

are you saying do the terms get closer to zero? yes.

Yes, this is what I was asking, but 'yes' is not the answer I had in mind!

Remember that

$$\left(1+\frac{1}{n}\right)^n \rightarrow e$$

!

 

[nuclearrape66]

and therefore by the divergence test...the series diverges?

 

[quasar987]

I was merely pointing out a sequence that behaves like the argument of your series and that does not go to 0.

However, you can cleverly manipulate the identity

$$\left(1+\frac{1}{n}\right)^n \rightarrow e$$

to find the precise value of the limit you're interested in.