Strategy for Testing Series, Infinite Series.

[mateomy]

I think I did this right...

<br /> \sum_{i=1}^{\infty} \frac{n}{e^{n^2}}<br />

I tried it with the root test to no avail. So I then tried it with the Ratio Test and I come to this expression...

<br /> \lim_{n \to \infty} \frac{(n+1)}{e^{2n} e(n)}<br />

...which is an indeterminate form (infinity over infinity)

so I take L'Hopital's Rule and come to...

<br /> \frac{1}{2e^{2n} (e)}<br />

Which, when taking the limit goes to zero to show absolute convergence.

Can someone confirm this for me?

Thanks!

 

[mateomy]

Dang. Thats supposed to be (in the original Series) e^(n^2)
It also follows that it should be e^(2n) below that.

 

[Dick]

Sure, that works. You are being little casual and throwing out the (n+1)/n part because it approaches 1 and not doing the full l'Hopital. But that's ok, because 1/(e^2n)*e definitely approaches 0.

 

[mateomy]

Awesome, thanks!