Ratio Test Questions/ Series Convergence

[calcboi]

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]

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)

 

[chisigma]

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]

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.

 

[alyafey22]

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.