Homework Help: Stuck with L' Hospital's Rule

1. Apr 13, 2012

EEintraining

1. The problem statement, all variables and given/known data
Ʃ n/e^n converge or diverge

2. Relevant equations

3. The attempt at a solution

I got this down to an improper integral using the integral test but I am weak at L'Hospitals rules and I was wondering if someone could help me out

I have

$\int n/e^n$ from 1 to infinity

down to

Limit b to infinity ne^n - e^n |from 1 to b

this gives me infinity - infinity so time for L'Hospitals ( forgive my spelling)
I know to divide by the recripical of either one but I get stuck from there

2. Apr 13, 2012

Dick

It's probably easier to use the ratio test. If you have to use the integral test you didn't get the integral quite right. There's a sign problem (which isn't terribly important) and don't you mean e^(-n) in the integral (which is terribly important)?

3. Apr 13, 2012

EEintraining

ok i will try the ratio test for this... I am studying for a test and have already worked this problem and turned it in when I got it back graded the only comments were that it was infinity-infinity and needs L'Hospitals rule. I had originally put infinity - infinity so it diverges. The original problem is correct it is n / e^n not e ^-n

4. Apr 13, 2012

Dick

No, I meant your integral should have been -n/e^n - 1/e^n or -ne^(-n) - e^(-n). It's not infinity-infinity.

5. Apr 13, 2012

EEintraining

Ok I did the Ratio test could you please check my work?

$\frac{n}{e^n}$

$\frac{n+1}{e^(n+1)}$ * $\frac{e^n}{n}$

so all e's cancel except 1 giving me

$\frac{n+1}{e*n}$ the limit of this is ∞/∞ with L'Hopitals i have

$\frac{1}{e}$ which is less then 1 so ratio test says converges

Last edited: Apr 13, 2012
6. Apr 13, 2012

Dick

Yes, the limit of the ratio is 1/e so it converges.

7. Apr 13, 2012

EEintraining

Awesome thanks... and I will try to work on form... that was the edited version you should have seen what i had first lol!