Does this series diverge or converge?
∑[(n=1),∞,(n)/(e^n )]

Question

Does this series diverge or converge?
∑[(n=1),∞,(n)/(e^n )]

anonymous · Answer

converge for sure

anonymous · Answer

How can you tell?  I mean, what test can be used to prove it? I've already tried the limit comparison test and direct comparison test but they don't seem to work.

anonymous · Answer

in fact it is rather small

anonymous · Answer

n<e^n so converges

anonymous · Answer

well i can tell with my eyeballs because n is a polynomial of degree 1 and e^n grows way faster than any polynomial but that doensn't help with your proof i suppose

anonymous · Answer

Oh right. This is a geometric series isn't it?

anonymous · Answer

you can use the ratio test if you like

anonymous · Answer

no it is not geometric

anonymous · Answer

Unfortunately, the lesson I'm currently doing hasn't covered it yet.

anonymous · Answer

hmm

anonymous · Answer

so only comparison test?

anonymous · Answer

zarkon, snappy way with comparison test?

anonymous · Answer

Well, I suppose I could just write the answer verbally....
As of now, we've learned the integral test, basic comparison, and limit comparison. Yeah.

zarkon · Answer

you can use the limit comparison with 1/n^2

though the ratio trest is the best(as you already noted)

zarkon · Answer

then use l'hospitals rule

zarkon · Answer

a few times

anonymous · Answer

oh well if you have integral test, it is an easy enough integral

anonymous · Answer

Oh, alright then. Thank you both very much!

anonymous · Answer

kinda odd to have to resort to an integral when a ratio would do, but what do i know?

zarkon · Answer

you know the taylor expansion of e^x?

anonymous · Answer

integration by parts vs laws of exponents...

anonymous · Answer

nope