User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)


 * $u \ v \ \mathsf{u} \ \mathsf{v} \ \nu \ \upsilon$

Anyone else have a hard time distinguishing between $u$ and $v$? I would like it to look more like this, does it confuse anyone else? It seems PW doesn't have the upgreek package. --GFauxPas 07:49, 27 January 2012 (EST)


 * Nope. Multiple years of extensive TeX writing and reading have trained my eye. I agree that referenced $v$ looks more distinguished, but imagine it is hard to implement. --Lord_Farin 08:08, 27 January 2012 (EST)

Derivative of a Series?
I'm learning convergence and divergence of sequences and series in my Calc Class, and Larson is implicitly using a theorem here, and it seems significant enough to warrant a page on PW if it's not already up, can someone explain what he did here?

The book is analyzing:


 * $\displaystyle \sum_{n=1}^{\infty} \ (-1)^n \frac {\sqrt{n}}{n + 1}$

He wants to use the Alternating Series Test, so to prove that $a_{n+1} \le a_n$, he creates a differentiable real function:


 * $f(x)= \displaystyle \frac {\sqrt{x}}{x + 1}$

and takes the derivative of that. The derivative is negative so $f$ is a decreasing function (for $x > 1$) but how does it prove that:


 * $\left \langle{ \displaystyle \frac {\sqrt{n}}{n + 1}}\right \rangle$

is decreasing? --GFauxPas 20:45, 4 February 2012 (EST)


 * Use FTIC and some result on inequality of integrals, together with the bounds $n, n+1$. I'm sure you'll figure from there. --Lord_Farin 17:14, 5 February 2012 (EST)


 * Oh, uh, hm. Does it use this guy?: Euler-Maclaurin Summation Formula. I don't know of any other things that tie together the limits of continuous functions with the limits of series. I guess I'll sleep on it. --GFauxPas 18:06, 5 February 2012 (EST)

PW pages for x grows faster than y
Do we have pages along the lines of:

for sufficiently large x, $e^x > x^n$, $x! > e^x$? Closest thing I've found was Powers Drown Logarithms. If we had pages for that, I might be able to write proofs for the limits of real functions approaching infinity being infinite, which there are hardly any of on PW. --GFauxPas 11:15, 5 February 2012 (EST)

Oh, and I'd also hope there's a theorem for the intuitively plausible statement that if, for x large enough, f(x) > g(x) for all x, and g(x) approaches infinity as $x \to +\infty$, then f(x) also approaches infinity. --GFauxPas 12:17, 5 February 2012 (EST)


 * In Dutch, the last result is called 'Duwstelling', which translates to 'Push Theorem'. Not sure if it's up, but it exists. --Lord_Farin 17:14, 5 February 2012 (EST)


 * is the proof something like this?


 * By hypothesis, the slower-growing function has $\forall M_1 > 0 : \exists N_1 > 0$ etc.


 * For any $M_2$ for the faster growing function, choose $N_1$ for $N_2$? Somehow that doesn't seem like enough. --GFauxPas 18:00, 5 February 2012 (EST)