Sequence e^2/2^n, converge or diverge (easy)

[8point1]

Homework Statement

does the series e^n/2^n converge or diverge?

does the series 2^n/e^n converge or diverge?

The Attempt at a Solution

I took lim→∞ e^x/2^x and am getting ∞, so it diverges, right?

I also used L'Hopital's rule and got the same result.

My prof hinted that I was wrong... what am I missing?

Thanks!

 

[Dick]

Homework Statement

does the series e^n/2^n converge or diverge?

The Attempt at a Solution

I took lim→∞ e^x/2^x and am getting ∞, so it diverges, right? My prof hinted that I was wrong.

Thanks!

Yes, you are right. e/2 is greater than 1. The series diverges.

 

[8point1]

what about 2/e?

 

[Dick]

what about 2/e?

You tell me. What do you know about the geometric series r^n?

 

[checkitagain]

Homework Statement

does the > >series < <e^n/2^n...

does the > >series < <2^n/e^n...

8point1,

there is a distinction. Those expressions are the general terms
of the sequences and the series.

The summations of those terms are the respective series.

 

[8point1]

The sequence diverges, so the series must diverge also, correct?

 

[Dick]

The sequence diverges, so the series must diverge also, correct?

That's correct for (e/2)^n. (2/e)^n is different.