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

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Homework Help Overview

The discussion revolves around the convergence or divergence of the series involving the expressions e^n/2^n and 2^n/e^n. Participants are exploring the behavior of these series as n approaches infinity.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants attempt to analyze the limits of the series using L'Hopital's rule and question the implications of their findings. There is also discussion about the relationship between sequences and series, particularly regarding the convergence of the series based on the behavior of the sequences.

Discussion Status

The discussion is ongoing, with participants providing insights and questioning assumptions. Some guidance has been offered regarding the distinction between sequences and series, and the implications of their convergence properties are being explored.

Contextual Notes

There are hints from a professor suggesting that some participants may be misunderstanding the convergence criteria. The distinction between the general terms of sequences and the summation of those terms is also under discussion.

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Homework Statement


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

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

The Attempt at a Solution



I took [itex]lim→∞ e^x/2^x[/itex] 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!
 
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8point1 said:

Homework Statement


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

The Attempt at a Solution



I took [itex]lim→∞ e^x/2^x[/itex] 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.
 
what about 2/e?
 
8point1 said:
what about 2/e?

You tell me. What do you know about the geometric series r^n?
 
8point1 said:

Homework Statement


does the [itex]> >[/itex]series [itex]< <[/itex][itex]e^n/2^n...[/itex]

does the [itex]> >[/itex]series [itex]< <[/itex][itex]2^n/e^n...[/itex]

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.
 
The sequence diverges, so the series must diverge also, correct?
 
8point1 said:
The sequence diverges, so the series must diverge also, correct?

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

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