Recent content by Tsunoyukami

I'm having some difficulty with a problem from Boyce & DiPrima's Elementary Differential Equations and Boundary Value Problems, 9th Edition. The problem comes from Section 2.8: The Existence and Uniqueness Theorem and is part of a collection of problems intended to show that the sequence...

That's a great idea, Quantum Defect! Thanks for the input. I was asking the question with respect to physics. I'm aware that we can concoct situations in which the x- and y- components are not independent of one another (ie. coupled equations). The mathematical definition has been suggested...

Thanks again, Bandersnatch. Those suggestions are pretty typical of what I've come across. I don't have anyone asking me to explain this in detail right now, but I was just curious if there were any simple explanations (proofs) one could offer to students who were skeptical of the claim that...

Thanks - I'm familiar with such motion. Of course it's possible to construct more complex problems with more complex (interesting) solutions (motion), but I'm more interested in trying to explain the basic fundamental notion that what happens in each cardinal direction is independent of motion...

Thanks! I'm familiar with these concepts and I'm not sure why I didn't think of them in the first place! Is there another simple explanation one could offer when explaining this idea to students unfamiliar with linear algebra? I've been asked about this concept by several friends taking high...

Consider a simple textbook problem in two dimensional kinematics - say, a projectile motion problem. I know that the x- and y- components of motion are independent of one another but I don't understand why. I know this is true due to everyday observation - empirical evidence of this being the...

Thank you for your prompt reply!

I'm working through some problems from Stewart's Calulus, 6ed. and am having some difficulty with certain limit proofs. In particular, there is no definition provided for limits of the form: $$ \lim_{x \to - \infty} f(x) = L $$ One of the exercises is to come up with a formal definition...

Oh, I see so it would be something like this: Proof Let ##\epsilon>0## be given. We must find ##\delta > 0## such that ##0<|x-a|<\delta \implies |f(x)+g(x) -(L+M)|<\epsilon##. By applying the triangle inequality we can write ##|f(x)+g(x) - (L+M)| \leq |f(x)-L| + |g(x)-M|##. I will pause the...

Prove the Sum Rule for Limits $$\lim_{x\to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M$$ Proof Assume the following: $$\lim_{x \to a} f(x) = L, \space\lim_{x \to a} g(x) = M$$ Then, by definition ##\forall \epsilon_1 > 0, \exists \delta_1 > 0## such that...

Suppose ##\lim_{x \to a} f(x) = \infty## and ##\lim_{x \to a} g(x) = c## where ##c## is a real number. Prove ##\lim_{x \to a} \big( f(x)+g(x) \big) = \infty##. Proof Assume ##\lim_{x \to a} f(x) = \infty## and ##\lim_{x \to a} g(x) = c##. Then, by definition: 1) For every ##M > 0##...

After consulting my professor this is the route that he intended for us to take (since we had not discussed the least upper bound). It was a pretty long proof written out formally but I'm happy to know how to approach this problem now. Thanks for the help everyone!

Alright, so I know that ##L - \epsilon < a_{N} ≤ L## or ##L - \epsilon < a_{N}## so I can write ##L < a_{N} + \epsilon##. We also know that ##a_{i}## and ##a_{j}## are both less than or equal to ##L## and ##i<j## so I can write ##a_{i} ≤ a_{j} ≤ L < a_{N} + \epsilon## so I have the inequalities...

If ##\left\{ a_{n} \right\}## is monotone increasing and there exists ##M \in \Re## such that for every ##n \in N## ##a_{n} ≤ M## prove that ##\left\{ a_{n} \right\}## converges. (Hint: Use the Cauchy sequence property. Recall: 1) ##\left\{ a_{n} \right\}## is Cauchy if and only if...

Thanks so much for the hint! So we can write: ##3^{3(k+1)+1}+2^{(k+1)+1}## ##= 27 \cdot 3^{3k+1}+2\cdot 2^{k+1}## ##=(25+2)\cdot 3^{3k+1}+2 \cdot 2^{k+1}## ##=25 \cdot 3^{3k+1} + 2 \cdot 3^{3k+1} + 2\cdot 2^{k+1}## ##= 25 \cdot 3^{3k+1} + 2 \cdot (3^{3k+1} + 2^{k+1})## By assumption we know...