So I found this on the Wolfram website today:
So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed:
- $ \lim_{x \to +\infty} \dfrac {\sqrt{x^3-x^2+3x}}{\sqrt{x^3}-\sqrt{x^2}+\sqrt{3x}} $
- $ \displaystyle\sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1} $
- $ \dfrac {\mathrm{d}}{\mathrm{d}u} \left[ \dfrac {u^{n+1}}{(n+1)^2} \cdot \left[ (n+1) \ln u - 1 \right] \right] $
- $ \displaystyle\int_0^{2\pi}\displaystyle\int_0^{\frac{\pi}{4}}\displaystyle\int_0^4 \left( \rho \cos \phi \right) \rho^2 \sin \phi \, \mathrm{d}\rho \mathrm{d}\phi \mathrm{d}\theta$
Ideas
- The degree of the numerator is $\frac{3}{2}$. The degree of the denominator is $\frac{3}{2}$. The expression is of an indeterminate form, namely $\frac{\infty}{\infty}$, so we use l'Hoptial's Rule.
No ideas, really.
Derivatives are always pretty easy, although this one is a bit bashy. Basically bash Product and Chain Rules, etc.
Integration by Parts bash? The $\mathrm{d}\theta$ part is very trivial. I mean for the $\rho$ and $\phi$.