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: