1. ## Converges or Diverges?

Need to know if this converges or diverges. Could someone show me the steps ??

2. ## Re: Converges or Diverges?

using the ratio test

$L=\lim \limits_{n\to \infty}~\dfrac{\dfrac{e^{n+1}+(n+1)}{(n+1)^5}}{ \dfrac{e^n+n}{n^5}} =$

$L=\lim \limits_{n\to \infty}~\dfrac{n^5(e^{n+1}+n+1)}{(n+1)^5(e^n+n)}$

$L = e > 1$ and thus the series is divergent

3. ## Re: Converges or Diverges?

I'm sorry I'm really bad at simplifying how did it give you e at the end ? I tried to simplify it and it never gives me e

4. ## Re: Converges or Diverges?

Originally Posted by steelmaste
I'm sorry I'm really bad at simplifying how did it give you e at the end ? I tried to simplify it and it never gives me e
should be pretty clear that $\dfrac{n^5}{(n+1)^5}$ goes to 1

should also be pretty clear that the $e^n$ term leaves the $n$ and constant terms in the dust so this becomes $\dfrac{e^{n+1}}{e^n} = e$

you can show all that formally but that's the common sense explanation.

5. ## Re: Converges or Diverges?

Originally Posted by steelmaste

Need to know if this converges or diverges. Could someone show me the steps ??
I offer another approach that uses the comparison & root test.
$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{e^n} + n}}{{{n^5}}}} \ge \sum\limits_{n = 1}^\infty {\frac{{{e^n}}}{{{n^5}}}}$
But
$\displaystyle \sqrt[n]{{\frac{{{e^n}}}{{{n^5}}}}} = \frac{e}{{{{\left( {\sqrt[n]{n}} \right)}^5}}} \to e > 1$
This shows divergence.

6. ## Re: Converges or Diverges?

Thank you ! this approach seems a lot more simpler, however I fail to understand how to make the bottom part (nsquarerootn)^5 go to 1 for it to be e/1. any tricks to simplify I could use? I pretty much tried to make it (n^(1/n))^5 but still dont see how it goes to 1, since my teacher said we cant just assume that infinity^(0) gives 1.

7. ## Re: Converges or Diverges?

Originally Posted by steelmaste
Thank you ! this approach seems a lot more simpler, however I fail to understand how to make the bottom part (nsquarerootn)^5 go to 1 for it to be e/1. any tricks to simplify I could use? I pretty much tried to make it (n^(1/n))^5 but still dont see how it goes to 1, since my teacher said we cant just assume that infinity^(0) gives 1.
If you do not know that $\displaystyle \sqrt[n]{n} \to 1$ then you are not able to use the root test.

8. ## Re: Converges or Diverges?

Originally Posted by steelmaste
Thank you ! this approach seems a lot more simpler, however I fail to understand how to make the bottom part (nsquarerootn)^5 go to 1 for it to be e/1. any tricks to simplify I could use? I pretty much tried to make it (n^(1/n))^5 but still dont see how it goes to 1, since my teacher said we cant just assume that infinity^(0) gives 1.
Let $y = n^{\frac 1 n}$. Then $\ln y =\frac{\ln n}{n} \to 0$. So $y\to 1$.