Does the series Ʃ n/e^n converge using the integral test?

[EEintraining]

Homework Statement

Ʃ n/e^n converge or diverge

Homework Equations

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

 

[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)?

 

[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

 

[Dick]

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

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

 

[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'hospital's i have

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

 

[Dick]

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

 

[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!