Recent content by Ethan Godden

Thank you, I understand now. It kind of odd that I haven't had problems before with this misunderstanding.

Ok, what about a function f:ℝ→ℝ such that f(x)=√x? This would mean its not defined for x<0, but how could we deduce that from "f: ℝ→ℝ" only like in this question? I think I found a reason for why s(p) is defined for every p. I believe part 3 of the axiom implies s∈ℕ→s(p)∈ℕ because given a set...

I understand that if s(p) exists, its image lies in ℕ by the axiom, but isn't it possible that s(p) is not defined at p? After all, doesn't the axiom say that the function s is injective. Wouldn't it need to be bijective or surjective in order for s(p) to be defined for all possible p ∈ ℕ? I...

Homework Statement This theorem comes from the book "The Real Numbers and Real Analysis" by Bloch. I am having a hard time understanding a particular part of the proof given in the book. Prove the following theorem: There is a unique binary operation +:ℕ×ℕ→ℕ that satisfies the following two...

Homework Statement Explain how a cell can use catabolic reactions to drive anabolic reactions, despite energy loss in the form of entropy and heat. Homework Equations Catabolic Pathway: A series of reactions that results in the breakdown of larger,more-complex molcules into small, less complex...

Okay, you used the squeeze theorem which makes sense, but why doesn't the test for divergence work? Isn't (-1)∞ undefined meaning the limit is undefined meaning the series is divergent?

Your work also looks good to me. The one thing I noticed is the answer from the book would be right if you take the negative away from the 1. I'm guessing the book has a typo.

Homework Statement I am supposed to determine whether the summation attached is convergent or divergent Homework Equations Alternating Series Test Test for Divergence The Attempt at a Solution The attempted solution is attached. Using the two different tests I am getting two different answers.

Homework Statement The problem is attached Homework Equations ΔEint=Q+W W=-PΔV The Attempt at a Solution Attempted solution is in the attachment. The problem is I am not getting the same answer as the supposed correct answer. Thank You, Ethan

I guess my main question is why are you dealing with sec(u) when the question ends up with sin(u)/a2. Shouldn't I be drawing a trig ciricle for sin(u) and not sec(u)?

Thank you, My only follow up question is why is this significant? I am not at sin(u)/a. I know by the trig circle that the opposite side is equal to x, the adjacent side is a, and the hypotenuse is √(x2+a2 ). This means sin(u)=x/√(x2+a2 ). I don't know where the sec(u) back substitution came...

how do I know sec(x)< 0? The only information given is the integral at the beginning and that a>0.

The one idea I have, by the way, is when I simplify the √sec2(x), it could become -sec(x). This is my only idea.

Homework Statement The problem is the integral attached Homework Equations sec2(u)=(1+tan2(x)) a2+b2=c2 ∫cos(u)=-sin(u)+C The Attempt at a Solution The solution is attached. I am wondering if someone could give me a hint where I went drastically wrong or where I possibly dropped a negative...

The mass of the system decreases in half for each spring system while the spring constant remains constant. This means each spring has a frequency √2 less than the spring with two masses on each end. But the two springs with one mass is the same as the original scenario so the new two mass...