Back before my now-ended blogging hiatus, the server machinery that keeps ScienceBlogs running was not so snazzy as it is now. Now it’s running a WordPress implementation that includes LaTeX support. LaTeX is a free environment for (among other things) typesetting mathematics. Let’s give it a test run:

$latex 3 = \sqrt{6+\sqrt{6+\sqrt{6+\cdots}}} &s=2$

I actually came across the above expression in a Mathematica book, which used it as an example of how Mathematica could do typesetting. It didn’t include anything other than the expression – no proof or anything else. We might as well test it out to see if it’s true. First, take a pocket calculator and start with the number 6. Take the square root and get (2.44949…). Add 6, and get (8.44949…). Take the square root and get (2.9068…). Add 6 and take the square root and get 2.98443…

Keep going and it sure looks like you’re closing in on 3, which is a good sign. But that’s not a proof. We could make our train of thought more systematic. We’re talking about a sequence of numbers $latex S_n$, where each $latex S_n$ in the sequence is given by the definition:

$latex S_{1} = \sqrt{6} &s=2$

$latex S_{n+1} = \sqrt{6 + S_n} &s=2$

Well, *if* this sequence converges, we’re just saying that each successive $latex S_{n}$ starts to close in on the final result $latex S$.

$latex \displaystyle \lim_{n \to \infty}S_{n} = S &s=2$

So the defining relationship becomes:

$latex S = \sqrt{6 + S} &s=2$

Which we can solve by inspection (or by algebra, if you prefer): $latex S = 3$.

Now notice that I said this logic was valid if and only if the sequence converges. Does it? Yes, but I’ll leave proving it as a challenge for people who’re looking for Calc 2 practice problems.

Hey, that wasn’t bad. This post took me massively less time than it would have back when I had to generate each equation image separately and upload it to the website – especially if I discovered a typo. I’m not sure if LaTeX works in comments, but if you’d like to give it a shot the WordPress-specific syntax is here. (Also for new readers: I put math posts in the Physical Science category because ScienceBlogs doesn’t have a dedicated math category.)