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

  • Thread starter nuclearrape66
  • Start date
  • Tags
    Series
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.
  • #1
nuclearrape66
31
0
does the series from 1 to infinity of [tex]\sum[/tex](1 - (2/n))^3n convegre or diverge.

i've tried almost all the tests and can't figure it out...
 
Physics news on Phys.org
  • #2
FAQ: Why hasn't anybody answered my question? - https://www.physicsforums.com/showthread.php?t=94383
 
  • #3
Have you even tried the simplest test of them all? Does the argument of the series goes to 0 as n increases without bounds?
 
  • #4
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.
 
  • #5
if you can hint me to what test i should use maybe i can figure it out.
 
  • #6
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

[tex]\left(1+\frac{1}{n}\right)^n \rightarrow e[/tex]

!
 
  • #7
and therefore by the divergence test...the series diverges?
 
  • #8
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

[tex]\left(1+\frac{1}{n}\right)^n \rightarrow e[/tex]

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

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

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.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
215
  • Calculus and Beyond Homework Help
Replies
2
Views
431
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
433
  • Calculus and Beyond Homework Help
Replies
1
Views
499
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
894
  • Calculus and Beyond Homework Help
Replies
2
Views
891
  • Calculus and Beyond Homework Help
Replies
6
Views
370
Replies
29
Views
4K
Back
Top