Proving \lim_{n\rightarrow\infty} \frac{p(n)}{e^n} = 0

[Zhalfirin88]

Homework Statement

Prove \lim_{n\rightarrow\infty} \frac{p(n)}{e^n} = 0 where p(x) = a_k x^k + ... + a_1 x + a_0 (with real coefficients a_i in \mathbb{R} )

The Attempt at a Solution

I thought about using series to try and prove this, but I couldn't get it to work out and I think there is an easier way.

\frac{p(n)}{e^n} = \frac{ \sum_{n=0}^\infty a_k n^k}{\sum_{n=0}^\infty \frac{n^k}{k!}} = \frac{ \sum_{n=0}^\infty a_k}{\sum_{n=0}^\infty \frac{1}{k!}}

 

[LCKurtz]

Homework Statement

Prove \lim_{n\rightarrow\infty} \frac{p(n)}{e^n} = 0 where p(x) = a_k x^k + ... + a_1 x + a_0 (with real coefficients a_i in \mathbb{R} )

The Attempt at a Solution

I thought about using series to try and prove this, but I couldn't get it to work out and I think there is an easier way.

\frac{p(n)}{e^n} = \frac{ \sum_{n=0}^\infty a_k n^k}{\sum_{n=0}^\infty \frac{n^k}{k!}} = \frac{ \sum_{n=0}^\infty a_k}{\sum_{n=0}^\infty \frac{1}{k!}}

Replace n by a continuous variable x and use L'Hospital's rule.

 

[Zhalfirin88]

I don't think I'm allowed to use L'Hospital's rule, we have not proven it in class yet.

 

[Robert1986]

Could you show that a_kn^k/e^n goes to 0 as n goes to infinity? Then each lower order term would go to 0, as well. What tests/methods are you allowed to use? Does this have to be strict \epsilon - \delta from the defimition of limit?

 

[Zhalfirin88]

No we are proving things based on the definition of convergence, for instance, our proofs start off as "Let \epsilon > 0 be arbitrary, and choose N > ... such that for all n > N ... etc. We haven't done epsilon delta proofs yet.

 

[Robert1986]

Whoops! yes, \epsilon - N is what I meant. But anyway, could you show that a_kn^k/e^n goes to 0? For instance, could you do something like this:
Let a = sup \{|a_i| : 0 \leq i \leq k \} then a_kn^k/e^n \leq an^k/e^n and then show that an^k/e^n goes to 0? Then deduce that the lower order terms go to 0? Of course, you are going to have to take more care than I did because 1)you'll need to find an N and 2)you'll need to take care of the times when you have negative a_k's. Anyway, am I making sense?

 

[Zhalfirin88]

Actually, since a_kn^k/e^n \leq an^k/e^n is the largest term of the polynomial, couldn't you just find a N that suits this expression? Because p(n)/e^n can be written as \frac{a_0}{e^n} + \frac{a_1 n}{e^n} + ... + \frac{a_k n^k}{e^n}, and since you choose n > N always, then the N that you pick for the largest term will be true for all terms, which will satisfy the definition, correct?

 

[Robert1986]

Correct; that's what I had in mind. Now, find that N. Is this an Analysis class? I had terrible trouble proving things like this because it is so freaking hard to know what you are allowed to use. For example, L'Hospital's is off the table, but what can you use? The obvious answer, at first, is that you can only use stuff that you have already learned in the analysis class but I'd be willing to bet you haven't defined e^x, yet, is this correct? Do you find this a confusing as I did?

 

[Zhalfirin88]

Yes, an analysis class, but yeah it's a little awkward, especially since the professor used a derivative to prove some lemma or something, then goes and says "we shouldn't be using derivatives yet, but I'll use it anyways" :) And no we have not defined e^x yet, correct