Recent content by tainted

Alright thanks man, that's all I needed, I got it!

Homework Statement Ok, I have this problem this week. (1) Consider\ a\ tiling\ of\ the\ unit\ sphere\ in\ \mathbb{R}^{3}\ by\ N\ equilateral\\ triangles\ so\ that\ the\ triangles\ meet\ full\ edge\ to\ full\ edge\ (and\ vertex\ to\ vertex).\\ Show\ that\ the\ only\ possibilities\ for\ N\...

\displaystyle\ \ \left.\int_{0}^{\infty} x^{n+1}e^{-x} dx= -{\LARGE{(}}x^{n+1}e^{-x}\right|_0^\infty{\LARGE{)}} + (n+1)\int_0^\infty e^{-x}x^{n} dx\ \\ = {\LARGE{[}}-\frac{x^{n+1}}{e^{\infty}} + x^{(n+1)}{\LARGE{]}} + (n+1)n! \\ = {\LARGE{[}}x^{(n+1)}{\LARGE{]}} + (n+1)n! \\ = (n+1)n! +...

mmk, but what would the limit approach? \lim_{x \to \ ?} \frac{-x^{(n+1)}}{e^{x}} where f(x) = -x^{(n+1)}\\ g(x) = e^{x}

-x^{n+1}e^{-x} + (n+1)n! So do I have to prove the first part equals 0?

Yeah, ha I'm posting as I work on other problems, so I post a little bit, and then I update. It should be edited as far as I have gotten now. I'm not really sure what to do to prove that equals (n+1)n!

\int_{0}^{\infty} x^{n+1}e^{-x} dx,\ \\ u = x^{n+1} \\ du = (n+1)x^{n} dx\ \\ dv = e^{-x} dx \\ \frac{dv}{dx}\ = e^{-x} \\ v = -e^{-x}\\ \int udv\ = uv - \int vdu\ \\ -x^{n+1}e^{-x} + (n+1)\int e^{-x}x^{n} dx\

Thanks, yeah I had that written, but I didn't get it down, can you tell me what was wrong with my LaTeX before I continue the rest of my work?

Thanks guys! I looked up integration by parts, and got to \int_{0}^{\infty} x^{n+1}e^{-x} dx,\ \\ u = x^{n+1} \\ du = (n+1)x^{n} dx\ \\ dv = e^{-x} dx \\ \frac{dv}{dx}\ = e^{-x} \\ v = -e^{-x}

Well I would assume it is integrating the parts that are multiplied together, but I do not know how to do that.

Homework Statement Prove \int_{0}^{\infty}x^{n}e^{-x} dx = n!Homework Equations 0! = 1 (by convention)The Attempt at a Solution Basic step: n=0 \\ \int_{0}^{\infty}x^{0}e^{-x} dx\ = 0! = 1\\ \int_{0}^{\infty}e^{-x} dx\ = -[e^{-\infty}-e^{0}]\\ -[e^{-\infty}-e^{0}] = -[\frac{1}{e^{\infty}}-1]\\...

Ok thanks guys! I think I got it, could you tell me if my attempted answer to the question appears to be sufficient?

Ok well of course -sin(x) = sin(-x) but does that hold true when there is a constant in front of x?

Homework Statement Prove the following formula \int_{-\pi}^{\pi} \sin(mx)\cos(nx)\,dx = 0\\ (m, n = \pm 1, \pm 2, \pm 3, ...) Homework Equations \sin(A)\cos(B) = \frac{1}{2}[\sin(A-B)+\sin(A+B)] The Attempt at a Solution \int_{-\pi}^{\pi} \sin(mx)\cos(nx)\,dx\\ \int_{-\pi}^{\pi}...

Homework Statement Consider a tiling of the unit sphere in ##\mathbb R^3## by equilateral triangles so that the triangles meet full edge to full edge (and vertex to vertex). Suppose n such triangles meet an one vertex. Show that the only possibilities for n are ## n=3 ##, ##n = 4##, or...