Limit of n[1 - exp(ia/n)] as n approaches infinity

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Homework Help Overview

The discussion revolves around finding the limit of the expression n[1 - exp(ia/n)] as n approaches infinity, where a is a fixed real number. Participants explore various approaches to evaluate this limit without relying solely on l'Hôpital's rule.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the validity of using l'Hôpital's rule and question whether the expression should involve exp(ia/n) or exp(a/n). There are considerations of oscillation in the real and imaginary parts of the expression as n approaches infinity. Some participants suggest expanding the exponential or cosine functions as power series, while others express uncertainty about the effectiveness of these expansions without derivatives.

Discussion Status

The discussion is active, with participants offering various insights and suggestions for evaluating the limit. There is a mix of opinions regarding the use of series expansions and l'Hôpital's rule, with some participants expressing confusion about the implications of different approaches. No consensus has been reached, but several productive lines of reasoning are being explored.

Contextual Notes

Participants note a preference to avoid methods that rely on derivatives or series expansions, indicating a desire to explore alternative approaches. There is also mention of the potential for misinterpretation of the limit's form, which could complicate the evaluation process.

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[SOLVED] Limit n[1 - exp(ia/n)]

Homework Statement
What is the limit of n[1 - exp(ia/n)] as n goes to infinity (a is fixed real number).

The attempt at a solution
n[1 - exp(ia/n)] = n / [1 - exp(ia/n)]-1 and by using l'Hosptal's rule, I get that the limit diverges. Now how can I do this without relying on l'Hospital's rule?
 
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Hi e(ho0n3! :smile:

Are you sure you mean n[1 - exp(ia/n)], and not n[1 - exp(a/n)]?

That's n(1 - cos(a/n)) - i sin(a/n).

The real part oscillates between … ?, and the imaginary part oscillates between … ? :smile:
 
tiny-tim said:
Are you sure you mean n[1 - exp(ia/n)], and not n[1 - exp(a/n)]?
No.

That's n(1 - cos(a/n)) - i sin(a/n).
That's correct.

The real part oscillates between … ?, and the imaginary part oscillates between … ? :smile:
I don't think it oscillates because a/n -> 0 as n -> infinity. 1 - exp(ia/n) -> 0 as n -> infinity. Or am I wrong?
 
oops!

e(ho0n3 said:
I don't think it oscillates because a/n -> 0 as n -> infinity. 1 - exp(ia/n) -> 0 as n -> infinity. Or am I wrong?

oh, sorry, I wasn't thinking straight. :redface:

Yes, so just expand exp(ia/n), or cos(a/n), as powers of a/n … :smile:
 
If I'm going to expand cos(a/n) as a power series, I might as well use l'Hospital's rule don't you think? And I don't see how this would help anyways.
 
e(ho0n3 said:
If I'm going to expand cos(a/n) as a power series, I might as well use l'Hospital's rule don't you think? And I don't see how this would help anyways.

ah, but you only need the first couple of terms of the expansion (for either cos or exp)! :smile:

and where's the fraction that you'd apply l'Hôpital's rule (that's the official spelling!) to?

Alternatively, if you prefer, you could rewrite 1 - cos(a/n) in terms of sin(a/2n).
 
tiny-tim said:
ah, but you only need the first couple of terms of the expansion (for either cos or exp)! :smile:

I know that but I don't understand how it is going to work. For example, the expansion of cos(z) about a/n is cos(a/n) + ... where ... are terms containing powers of z - a/n. Evaluating this at z = a/n gives me cos(a/n) again.

Please note that I don't want to use any method that relies on computing derivatives.

and where's the fraction that you'd apply l'Hôpital's rule (that's the official spelling!) to?

It's in my first post: n / [1 - exp(ia/n)]-1. (http://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital" states he spelled his name with an s so I'm just following the man's preference.)

Alternatively, if you prefer, you could rewrite 1 - cos(a/n) in terms of sin(a/2n).
I'm not familiar with this identity. Would you care to explain?
 
Last edited by a moderator:
e(ho0n3 said:
Homework Statement
What is the limit of n[1 - exp(ia/n)] as n goes to infinity (a is fixed real number).

The attempt at a solution
n[1 - exp(ia/n)] = n / [1 - exp(ia/n)]-1 and by using l'Hosptal's rule, I get that the limit diverges. Now how can I do this without relying on l'Hospital's rule?

I can't think of any way to do this that doesn't involve l'Hopital or derivatives (or series expansions, which are pretty much the same thing). But l'Hopital doesn't show that the series diverges. I will admit that the infinity/infinity form that you picked seems to just make the problem harder. Try (1-exp(ia/n))/(1/n).
 
Hi e(ho0n3! :smile:

The expansion for cosθ is 1 - θ²/2 + θ^4/4! - θ^6/6! + …

(and the expansion for sinθ is θ - θ^3/3! + θ^5/5! - θ^7/7! + …)
Wikipedia states he spelled his name with an s so I'm just following the man's preference.

Well, I never knew that! :rolleyes:
I'm not familiar with this identity. Would you care to explain?

Sure … you need to learn (though not necessarily for this question):
sin2θ = 2.sinθ.cosθ;
cos2θ = cos²θ - sin²θ;
1 + cos2θ = 2cos²θ;
1 - cos2θ = 2sin²θ;​
put θ = a/2n, and it's the last one. :smile:

(btw, if you did want to use l'Hôpital's rule, remember it's really only for 0/0, not for ∞/∞, so follow Dick's advice. :smile:)
 
  • #10
tiny-tim said:
(btw, if you did want to use l'Hôpital's rule, remember it's really only for 0/0, not for ∞/∞, so follow Dick's advice.

l'Hopital can be used for either 0/0 or infinity/infinity forms. The only issue is that sometimes one form can lead to an easier solution than the other. Choosing the wrong form can sometimes complicate the problem rather than simplify it. If things don't seem to be going well, try and rearrange the indeterminant form to make life easier.
 
  • #11
OK. I will satisfy myself with the derivation using l'Hospital's rule. Thank you tiny-tim and Dick.
 

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