Formal proof of sequence limits

1John5vs7

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I've got all of the homework from Stewart's "Calculus: Concepts and Contexts" section 8.1 done, but I'm stuck on two additional brainbusters the professor assigned (without, I might add, giving us any formal education in formal proof writing...frustration).

Here they are:

#1. Prove that lim as n-->inifinity of 1/(n^2) = 0.

I know intuitively that as n gets bigger and bigger, 1/n^2 gets smaller and smaller because the denominator is growing while the numerator remains constant. Thus, as n goes to infinity, the value of 1/n^2 goes to 0. But that isn't a FORMAL proof. I'm assuming she's looking for an epsilon proof, but I don't really understand how to go about setting that up.

I wrote a proof that a limit must exist based on the fact that we can demonstrably pick an epsilon as arbitrarily small as we like but there is still an N such that all n's greater than that N result in a value of 1/(n^2) < epsilon.

But all that does for me is show that SOME limit exists. It doesn't show that the limit is zero. How would I go about showing that it's zero?






#2. Prove that (n/(n+1))^n = 1/e

For this one, she gave us the "shorter method" in class that looks like this:

First she broke this out to (1 + x/n)^n and then took Euler's number raised to the ln of this value: e^ln(1+(x/n))^n.

This let her bring the n exponent down in front of ln, thus canceling Euler's and resulting in nln|1+ x/n|. Since this results in infinity * 0 and that's an indeterminate form, she rewrote it thus:

(ln|1+ x/n|)/ 1/n and then applied l'Hôpital's rule rendering it in a fashion that would be way too messy to type out but basically resulted in the whole mess being equivalent to 1/e.

Anyway, she said that was the "short method" and we should use the "longer method" to prove it. Those were literally the only instructions she gave the class. I have begun to think perhaps I am on the receiving end of the perfect storm of: a.) not knowing quite enough calculus to understand how to do this, b.) having a professor who doesn't know quite enough about didactic method to be an effective educator, and c.) participating in a summer intensive that doesn't allow quite enough time to complete homework sets given the lack of office hours and no TA or tutor to assist. I am tragically adrift in this unholy confluence. God save me!

If anyone can help with either problem, you'd be an absolute brick, a true chum. Many thanks.
 
I've got all of the homework from Stewart's "Calculus: Concepts and Contexts" section 8.1 done, but I'm stuck on two additional brainbusters the professor assigned (without, I might add, giving us any formal education in formal proof writing...frustration).

Here they are:

#1. Prove that lim as n-->inifinity of 1/(n^2) = 0.

I know intuitively that as n gets bigger and bigger, 1/n^2 gets smaller and smaller because the denominator is growing while the numerator remains constant. Thus, as n goes to infinity, the value of 1/n^2 goes to 0. But that isn't a FORMAL proof. I'm assuming she's looking for an epsilon proof, but I don't really understand how to go about setting that up.

I wrote a proof that a limit must exist based on the fact that we can demonstrably pick an epsilon as arbitrarily small as we like but there is still an N such that all n's greater than that N result in a value of 1/(n^2) < epsilon.

But all that does for me is show that SOME limit exists. It doesn't show that the limit is zero. How would I go about showing that it's zero?






#2. Prove that (n/(n+1))^n = 1/e

For this one, she gave us the "shorter method" in class that looks like this:

First she broke this out to (1 + x/n)^n and then took Euler's number raised to the ln of this value: e^ln(1+(x/n))^n.

This let her bring the n exponent down in front of ln, thus canceling Euler's and resulting in nln|1+ x/n|. Since this results in infinity * 0 and that's an indeterminate form, she rewrote it thus:

(ln|1+ x/n|)/ 1/n and then applied l'Hôpital's rule rendering it in a fashion that would be way too messy to type out but basically resulted in the whole mess being equivalent to 1/e.

Anyway, she said that was the "short method" and we should use the "longer method" to prove it. Those were literally the only instructions she gave the class. I have begun to think perhaps I am on the receiving end of the perfect storm of: a.) not knowing quite enough calculus to understand how to do this, b.) having a professor who doesn't know quite enough about didactic method to be an effective educator, and c.) participating in a summer intensive that doesn't allow quite enough time to complete homework sets given the lack of office hours and no TA or tutor to assist. I am tragically adrift in this unholy confluence. God save me!

If anyone can help with either problem, you'd be an absolute brick, a true chum. Many thanks.

For the first one, what is the formal definition you use. It should be something like the following:
If the limit of f(n)=a as n goes to infinity then for all \(\displaystyle \epsilon\) greater than zero, there exist an N such that if n>N, |f(n) - a|<\(\displaystyle \(\displaystyle \epsilon\)\).

Now note than if n>N, then 1/n2=|1/n2|<1/N2 and that for any \(\displaystyle \epsilon\)>0, that
if N > \(\displaystyle \sqrt{ceiling(\frac{1}{\epsilon}}\)), \(\displaystyle \epsilon\)>1/max(N2, 4)

I'll have to wait on the second one as I need to do something else right now.
 
I strongly empathize with your situation. I too sometimes look at my calculus book and go "What am I looking at? This makes no sense!" I think I can help you with the first problem.

In the problem, f(x) = x-2 with a given limit of 0, so the statement you want to prove is: \(\displaystyle \text{if} \, x \in (N,\infty) \, \text{then} \, \frac{1}{x^2} \in (0-\epsilon, 0+\epsilon)\).

That statement is equivalent to: \(\displaystyle \text{if} \, x > N \, \text{then} \, |\frac{1}{x^2} - 0| < \epsilon\)

And from that, you arrive at the logical conclusion that \(\displaystyle \text{if} \, x > N > 0 \, \text{then} \, \frac{1}{x^2} < \epsilon\)

From reading your post, that seems to be as far as you got on your own. Essentially, you now have a delta-epsilon proof, but instead of using the symbol delta, we're using the letter N. So, logically, the idea is to find a value for N in terms of epsilon. As a nudge in the right direction, think about what it means that x > N. If x > N, then x2 > N, right? See if you can continue along this line of thinking. You're close to the end of the proof.

For the second problem, I honestly have no idea. I assume the "longer method" your professor spoke of is using a delta-epsilon style proof for the limit. But limit proofs were never my strong point, and I haven't even the slightest clue how to prove a limit that complicated. Sorry.

My final point of advice is if you feel like you're "drowning" in this class, I'd talk to your teacher in person. See if you can snag her for a few minutes after class. She may not have official office hours, but I think that any teacher worth their salt would make time to help a struggling student.
 
Prove that (n/(n+1))^n = 1/e:
Personally, I would turn it upside down and use the fact that if lim[f(n)]=a and a is not zero, then lim[1/f(n)]=1/a.

I would then note that
\(\displaystyle (\frac{n+1}{n})^n = \frac{(n+1)^n}{n^n}\)
and remember that
(ax)' = ln(a) ax
 
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