23 July 2010

Harmonic series (further test in LaTeX)

I learned yesterday, through Evolutionblog, that I can write LaTeX as long as I put a little script at the bottom of the posting, which can be found here. This is my attempt to make it work.

Well, since the actual posting on Evolution blog was on harmonic series, let me write Pietro Mengoli's proof of its divergence. Recall that the harmonic series is
S = 1 + \frac{1}{2} + \frac{1}{3} + \cdots.
Let us prove that $S=\infty$. Mengoli did the following. He grouped all terms, except the first one, in triples:
S = 1 + \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right)
+ \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} \right)
+ \left(\frac{1}{8} + \frac{1}{9} + \frac{1}{10} \right) + \cdots
Then he observed that each triple is larger than three times the middle term:
\frac{1}{n-1} + \frac{1}{n} + \frac{1}{n+1} > \frac{3}{n}.
And so he wrote
S > 1 + \frac{3}{3} + \frac{3}{6} + \frac{3}{9} + \cdots = 1 + 3S.
Since no finite positive number can be larger than 3 times itself plus 1, he concluded that $S=\infty$.

To see that the inequality above is true write it as
\frac{1}{n-1} + \frac{1}{n+1} > \frac{2}{n},
which is equivalent to
\frac{2n}{n^2-1} > \frac{2}{n}
which is obviously true.

Another way to see the inequality (and more) is to observe that if $X$ is a positive random variable, which is not a constant, then $E(1/X) > 1/E(X)$. To see this, let $Y$ have the same distribution as $X$ but be independent of it. Since $X^2+Y^2 > 2 XY$ we have $2 < \frac{X}{Y} + \frac{Y}{X}$, and, by taking expectations, $2 < E(X) E(1/Y) + E(Y) E(1/X) = 2 E(X) E(1/X)$, as claimed. Then apply this to a random variable $X$ which takes values $n-1$ or $n$ or $n+1$, each with probability $1/3$. This gives Mengoli's inequality. Mengoli also showed that the alternating harmonic series converges to the natural logarithm of 2:
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots = \log 2.
Mengoli was born in 1626 in Bologna and died in 1686 in the same town. He also computed the sums
 \sum_{n=1}^\infty \frac{1}{n(n+k)},
for $k=1,2,3,\ldots$ and showed that the result is always a rational number. He naturally wondered what the sum equals to when $k=0$. This was the famous Basel problem, which he posed in 1644. It was shown by Leonhard Euler in 1735 that the sum, for $k=0$, equals $\pi^2/6$. It is not surprising that Mengoli could not find this.

I'm still not happy with this way of writing LaTeX. I can't figure out how to number equations. If only html and LaTeX were fully compatible...

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant