Wednesday, May 27, 2015

Evaluating the Alternating Harmonic Series

Here's a fun little note I just found while organizing my desk. I discovered this proof during a class on infinite series last summer. The professor mentioned the limit of the alternating harmonic series, then commented offhand that "this limit is true, but we cannot prove it yet." Never one to shy from a challenge, I decided to see whether he was right, and had roughly this proof scrawled on a piece of paper by the end of class. He suggested that I try to publish it in an undergrad journal, but that didn't pan out for various reasons. Nevertheless, it's a fun proof, and I wasn't able to find it anywhere online, so I thought I'd share it here.

The alternating harmonic series is defined as

which, of course, is just the harmonic series with the sign of every other term flipped.

It is well known that the harmonic series diverges. Slightly less well known is that the alternating harmonic series converges to a very clean limit -- strangely enough, it turns out to be ln 2. This fact, if it comes up, is usually proved just as a trivial consequence of the Euler-Mascheroni constant's most common identity. Not only is that proof beyond the reach of many undergrads, it's also so trivial as to border on inane. It's like using a sledgehammer to pound in a nail: of course it works, but is it really the best way? Someone pass me a hammer.

Consider the following series:

Sunday, May 24, 2015

Finding Files with Multiple Substrings

I'd like to share with you a challenge I ran into last night, a problem that's really difficult unless you take to the command line.

Some background: I play Go. The GoGoD game database, which I recently bought and downloaded, has somewhere on the order of 80,000 Go game records in it. These games are all in .sgf format, which is a simple plaintext format. Records of games from professionals of many different ranks are included. The rating scale for professionals starts at 1 dan (or "shodan") and goes up to 9 dan, the highest rating a player can achieve. Games between professionals are always interesting, but games between 9dan professionals are, in particular, packed with subtlety and value for the ambitious student of Go.

Any game from this database is worth studying, but with over eighty thousand games to choose from, it helps to be able to pick highlights, and what better way than by focusing on games between 9dans?

If we decide to take this route, we're immediately faced with a second question: sure, we can just open games at random until we find one between 9dans, but this is boring and slow. Are we doomed to tedium, or is there some way we could search this database of more than eighty thousand games, pick out only the games between 9dans, and copy these over to a special "elite database"?

To the average user, this might sound impossible. On the command line, it's almost trivial.