root test for absolute convergence

[sambellamy]

I am asked to use the root test to determine whether the following series is absolutely convergent:
Ʃ_n=2^∞ [(-2n)/(n+1)]⁵ⁿ

I have determined that with the root test, this simplifies to:

32 * lim n-> ∞ |n/(n+1)|⁵, having taken out the |-2|⁵ and bringing it to the front. I understand that lim n-> ∞ |n/(n+1)| converges to 1, but my question is does this converge to 32 or 1? I am assuming 1 since I know that taking the derivative is allowable to come up with the same limit ratio, however the root test states different outcomes for L=1 vs. L=32.

Which is the actual value of the limit?

Thanks!

 

[HallsofIvy]

I am asked to use the root test to determine whether the following series is absolutely convergent:
Ʃ_n=2^∞ [(-2n)/(n+1)]⁵ⁿ

I have determined that with the root test, this simplifies to:

32 * lim n-> ∞ |n/(n+1)|⁵, having taken out the |-2|⁵ and bringing it to the front. I understand that lim n-> ∞ |n/(n+1)| converges to 1, but my question is does this converge to 32 or 1? I am assuming 1 since I know that taking the derivative is allowable to come up with the same limit ratio, however the root test states different outcomes for L=1 vs. L=32.

Which is the actual value of the limit?

Thanks!

The limit "\(\displaystyle \lim_{n\to\infty}\left(\frac{n}{n+ 1}\right)^5\) is 1 but \(\displaystyle 32\lim_{n\to\infty}\left(\frac{n}{n+ 1}\right)^5\) is 32. Which are you asking about?

 

[sambellamy]

Thanks for your reply. I mean to ask, is the limit of 1 what i would use to determine to result of the root test? or is 32 the figure i would use for that? The text i am looking at spells out two different results for the limit being 1 and the limit being >1.

 

[Ishuda]

In order for
\(\displaystyle \Sigma_{n=0}^{n=\infty} a_n\)
to converge a_n must go to zero 'fast enough'. Now, for this problem,
\(\displaystyle a_n= (-2)^{5n} * (\frac {n + 1 - 1}{n+1})^{5n}= (-2)^{5n} * ( 1 - \frac {1}{n+1})^{5n}\)

Considering that
(1 + \(\displaystyle \frac{1}{n})^n \) --> e,
what can you say say about the limits of the a_n as n --> \(\displaystyle \infty\)

 

[Ishuda]

Thanks for your reply. I mean to ask, is the limit of 1 what i would use to determine to result of the root test? or is 32 the figure i would use for that? The text i am looking at spells out two different results for the limit being 1 and the limit being >1.

The root test is for the nth root of a_n or
\(\displaystyle [|\frac{-2n}{n+1}|^{5n}]^{\frac{1}{n}} = (\space \frac{2n}{n+1})^5 = 32\space (\frac{n}{n+1})^5 \)
so the 32 is part of that limit.