Recent content by moxy

Hmm, I had tried a proof by contradiction, but I hadn't gotten far. Your proof seems to make perfect sense, though. I'll have to chew it over and try to justify the logic of each step to myself before my assignment is due. Thank goodness I have the next few days off! Thank you so much for your...

Even just a general suggestion on how you might start trying to deconstruct a proof like this would be greatly appreciated.

Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...

You should never start a proof by assuming that what you're trying to prove is true. So you shouldn't do this: Claim: When x > 0, x+1/x ≥ 2 Proof: Let x>0. Assume x+1/x ≥ 2 ... If your TA wasn't paying attention and missed that you used if and only if statements, he might have thought that...

Since you used iff arrows, I don't see anything wrong with your proof. However, I think your TA just wanted you to start by writing something that you know is true (i.e. (x+1)2 ≥ 0) and then work towards what you're trying to prove. The way you did it makes it look like you started by assuming...

Oh, okay. The problem hint suggested I use L'Hopital, so I guess I was thinking "in the box" when it came to evaluating the limit. Thanks for the suggestions!

Okay, so that will help me show that the limit of the denominator is ∞, but then taking the derivative is going to be a huge pain...

Homework Statement ForQ(x) = x^k + \sum_{n=0}^{k-1} a_n x^n Find \lim_{x→∞}\left({Q(x)^{\frac{1}{k}} - x}\right) Homework Equations L'Hopital The Attempt at a Solution Q(x) - x^k = \sum_{n=0}^{k-1} a_n x^n Q(x) - x^k = (Q(x)^{\frac{1}{k}})^k - x^k = \left(Q(x)^{\frac{1}{k}} -...

\int_C \frac{1}{z^2 + 4} dz = \int_0^{2\pi} \frac{1}{(e^{iθ})^2 + 4} ie^{iθ}dθ Let θ = ∏t => dθ = ∏dt \int_0^{2\pi} \frac{1}{(e^{iθ})^2 + 4} ie^{iθ}dθ = \frac{1}{\pi}\int_0^{2} \frac{1}{(e^{i\pi t})^2 + 4} ie^{i\pi t}dt = \frac{i}{\pi}\int_0^{2} \frac{1}{1 + 4} e^{i\pi t}dt =...

Okay, that makes enough sense. Though I have a few other integration problems, namely ones from a section in the book before Cauchy's thm is mentioned. I guess if I did it incorrectly above, then I did all of the other ones wrong. Is there a method to actually calculate the integral?

Can I just use Cauchy's Theorem to say that since C is simple, closed, and rectifiable and f(z) is holomorphic in and on C the \int_c f(z) dz = 0? The book we use is very old and uses a lot of out of date terms...

Wait, if the problem was f(z) = 1/z, then I'd do the same as I did above and get, \int_c \frac{1}{z} dz= \int_0^{2\pi} \frac{1}{e^{iθ}} ie^{iθ} dθ = \int_0^{2\pi}i dθ= i \int_0^{2\pi}dθ = (i) θ\bigg|_0^{2 \pi} = i(2\pi - 0) = 2\pi i Should I be doing some substitutions, or finding the...

Homework Statement Given C is the unit circle, evaluate \int_C \frac{1}{z^2 + 4} dz Homework Equations unit circle: z = e^{iθ} The problem doesn't specify how many times to go around the unit circle or which way, so I'm going to assume once and counterclockwise. The Attempt at a...

Homework Statement "For what values of z does e^{e^x} = 1 ? If z_m and z_n range over distinct roots of this equation, is the set of distances d(z_m, z_n) bounded away from zero?" The Attempt at a Solution This equation doesn't have any solutions, does it? ew = 1 only when w = 0. w in...

Okay, that makes sense. But since I'm only comparing to other series, can I be sure that I'm catching all of the values of x where the original series converges? EDIT: Nevermind...since I'm comparing the original series based on specific intervals of x, I'll be fine. So I'll get the same...