Root Test to an Infinite Series

[Edder]

\(\displaystyle \sum_{n=1}^{\infty} \frac{3n + 2}{n} ({8/5})^n\)

The question asks me to compute the value of p for the series with the root test. I took the root test and got:

\(\displaystyle \sum_{n=1}^{\infty} \frac{3n +2}{n} ({8/5})\) , as with the limit of n going to infinity. By that notion, I got 3 x (8/5), which is 24/5, but the answer is incorrect. Can anyone tell what I am doing wrong?

Thanks

 

[HallsofIvy]

\(\displaystyle \sum_{n=1}^{\infty} \frac{3n + 2}{n} ({8/5})^n\)

The question asks me to compute the value of p for the series with the root test. I took the root test and got:

\(\displaystyle \sum_{n=1}^{\infty} \frac{3n +2}{n} ({8/5})\) , as with the limit of n going to infinity. By that notion, I got 3 x (8/5), which is 24/5, but the answer is incorrect. Can anyone tell what I am doing wrong?

Thanks

First that second sum has nothing to do with the root test- the root test says nothing about a sum of nth roots. Second, you did NOT take the nth root of the entire term.

The root test says "\(\displaystyle \sum_{n=1}^\infty a_n\) converges is \(\displaystyle lim_{n\to\infty} \sqrt[n]{a_n}< 1\) and diverges if that limit is greater than 1. Here, the nth term is \(\displaystyle \frac{3n+2}{n}(8/5)^n\). The nth root of that is \(\displaystyle \sqrt[n]{\frac{3n+2}{n}(8/5)^n}= \sqrt[n]{\frac{3n+2}{n}}(8/5)\). Now, what is the limit of \(\displaystyle \sqrt[n]{\frac{3n+2}{n}}\) as n goes to infinity?

 

[Edder]

First that second sum has nothing to do with the root test- the root test says nothing about a sum of nth roots. Second, you did NOT take the nth root of the entire term.

The root test says "\(\displaystyle \sum_{n=1}^\infty a_n\) converges is \(\displaystyle lim_{n\to\infty} \sqrt[n]{a_n}< 1\) and diverges if that limit is greater than 1. Here, the nth term is \(\displaystyle \frac{3n+2}{n}(8/5)^n\). The nth root of that is \(\displaystyle \sqrt[n]{\frac{3n+2}{n}(8/5)^n}= \sqrt[n]{\frac{3n+2}{n}}(8/5)\). Now, what is the limit of \(\displaystyle \sqrt[n]{\frac{3n+2}{n}}\) as n goes to infinity?

Yeah I forgot to take the nth root of both factors. I eventually realized that the answer was simply 8/5.

Thanks for the input though, appreciate it.