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Ratio Test Questions/ Series Convergence

calcboi

New member
Mar 31, 2013
16
I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
 

Petrus

Well-known member
Feb 21, 2013
739
Re: Ration Test Questions/ Series Convergence

I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
Hello,
When it says lim x->1 it means x goes to 1 but not actually equal to 1. We aproximite. I would like someone to Approve that what I say is correct. I am sure but when it comes to help other i somehow get uncertain

Edit: If you want less Then 1 I think you Will lim x->1 (negative way)
 
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chisigma

Well-known member
Feb 13, 2012
1,704
I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
The ratio test can be applied to the series with only positive or negative terms. Is Your series of this type?...

Kind regards


$\chi$ $\sigma$
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3)
As has already been said, the ratio test is not convenient here. The series is $\displaystyle\sum_{n=1}^{\infty}\frac{\cos n\pi}{n^{2/3}}=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{2/3}}$, so the series of the absolute values is divergent (Riemann's series with $p=2/3\leq 1$). On the other hand by Leibniz criterion, the series is convergent. This implies that the given series is conditionally convergent.
 
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ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Re: Ration Test Questions/ Series Convergence

Hello,
When it says lim x->1 it means x goes to 1 but not actually equal to 1. We aproximite.
It means we are choosing some \(\displaystyle |x-1|<\delta\) , which is a more general definition whether we are approaching from the right or the left. We can not say we are approximating the values of x since approximation has always a space of error.