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

• nuclearrape66
In summary, the question is whether the series from 1 to infinity of \sum(1 - (2/n))^3n converges or diverges. The person asking the question has tried various tests but is unable to come to a conclusion. The responder suggests using the divergence test, but also mentions that the argument of the series behaves like a sequence that does not go to 0. They suggest manipulating the identity of \left(1+\frac{1}{n}\right)^n to find the exact value of the limit.
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...

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

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

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.

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

nuclearrape66 said:
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

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

!

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

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.

## 1. What is the formula for the series $\sum (1 - \frac{2}{n})^3n$?

The formula for the series is $\sum (1 - \frac{2}{n})^3n = \frac{n}{n^3-6n^2+12n-8}$.

## 2. Does the series $\sum (1 - \frac{2}{n})^3n$ converge or diverge?

The series converges.

## 3. What is the limit of the series $\sum (1 - \frac{2}{n})^3n$ as n approaches infinity?

The limit of the series is 1.

## 4. What is the convergence test for the series $\sum (1 - \frac{2}{n})^3n$?

The convergence test for the series is the ratio test.

## 5. How many terms do you need to add to get an approximation of the series $\sum (1 - \frac{2}{n})^3n$ within a specific error margin?

The number of terms needed will depend on the error margin and can be calculated using the error bound theorem.

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