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...

Successful blogging

I guess this is why my blog is not too successful.

this is a test

$\int_{M} d \omega = \int_{\partial M} \omega$

New date for the end of the world: 21 May 2011

A certain simpleton, called Harold Camping, has "proved", using "mathematics", that the world will not end in 2012, as the Mayans, allegedly, predicted, and as the recent Hollywood movie proclaimed, but in 21 May 2011.

Here is how he did it:

• Jesus was crucified on 1 April 33.
• 2011-33 = 1978 years.
• 1978 x 365.2422 days (the number of days in each solar year) = 722,449.0716 days. That is, approximately, 722,500 days.
• But 722,500 = (5 x 10 x 17) x (5 x 10 x 17).
• And the Bible says (so says the above simpleton):
• 5 = Atonement
• 10 = Completeness
• 17 = Heaven

And then he said:

"I tell ya, I just about fell off my chair when I realized that: 5 times 10 times 17 is telling you a story, it's the story from the time Christ made payment for your sins until you're completely saved."

Why, he is absolutely right! Perfect line of reasoning. No flaws, no nothing. Just the truth, a heavenly one, that is.

We know that this time, Camping is right. Indeed, we can forgive his earlier mistake:

On Sept. 6, 1994, dozens of Camping's believers gathered inside Alameda's Veterans Memorial Building to await the return of Christ, an event Camping had promised for two years. Followers dressed children in their Sunday best and held Bibles open-faced toward heaven. But the world did not end. Camping allowed that he may have made a mathematical error. He spent the next decade running new calculations, as well as overseeing a media company that has grown significantly in size and reach.^[source]

And his efforts were redeemed: not only did Camping learn more mathematics (indeed, how could he ever come up with the complicated formulas above), not only did he read the Bible line by line, but he also asked for forgiveness for his previous error and is now, I suppose, preparing for the doomsday. Apparently he has more followers now.

Rick LaCasse, who attended the September 1994 service in Alameda, said that 15 years later, his faith in Camping has only strengthened.
"Evidently, he was wrong," LaCasse allowed, "but this time it is going to happen. There was some doubt last time, but we didn't have any proofs. This time we do."

Yeah! PROOFS! They have proofs. And so we must accept them. They are fullproof foolsproofs proofs.

"a Nadder!"'s Friday Links

The Friday Links from the blog of a Nadder are not to be missed, He does a great job and I'd like to encourage him by reposting here:

http://anadder.com/friday-links-9-oct-09
http://anadder.com/friday-links-4-dec-09
http://anadder.com/friday-links-12-feb-10
http://anadder.com/friday-links-26-feb-10
http://anadder.com/friday-links-9-apr-10
http://anadder.com/friday-links-23-apr-10
http://anadder.com/friday-links-7-may-10
http://anadder.com/friday-links-18-jun-10
http://anadder.com/friday-links-21-may-10
http://anadder.com/friday-links-4-jun-10
http://anadder.com/friday-links-16-jul-10