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.
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.