Convergence

Let’s dig a little deeper behind the solutions to Challenges #1 and #2. What on earth does it mean for an infinite expression to have a “value”? Well, as noted in the solution to Challenge #1, what we’re really talking about is the value of the expression at various “stopping points” along the way: what happens to these values as the stopping points get further and further along?

Let’s look again at the infinite continued fraction from Challenge #1. To make things easier to read (and so they won’t take up as much space), we’ll use a common notation for continued fractions:

[a,b,c,d,\dots] = a + \cfrac{1}{b + \cfrac{1}{c + \cfrac{1}{d + \ddots}}}

So, for example, the expression from Challenge #1 can be written as [1,1,1,1,\dots], and the second expression from Challenge #2 can be written as [1,3,1,3,\dots]. As another example, [2,1,3] (note there are no ellipses) means 2 + \frac{1}{1 + \frac{1}{3}} which is equal to 11/4.

Now, let’s analyze various “stopping points” along the way to the infinite expression [1,1,1,1,\dots], starting with 1, then 1 + \frac{1}{1}, then 1 + \frac{1}{1 + \frac{1}{1}}, and so on:

\begin{array}{rcl} [1] & = & 1 \\ \left [ 1,1 \right ] & = & 2 \\ \left [ 1,1,1 \right ] & = & 3/2 = 1.5 \\ \left [ 1,1,1,1 \right ] & = & 5/3 \approx 1.66667 \\ \left [ 1,1,1,1,1 \right ] & = & 8/5 = 1.6 \\ \left [ 1,1,1,1,1,1 \right ] & = & 13/8 = 1.625 \\ \left [ 1,1,1,1,1,1,1 \right ] & = & 21/13 \approx 1.61538 \\ \vdots & = & \vdots \end{array}

(Try confirming these values for yourself.) As you can see, it seems like these numbers are getting closer and closer to something… here’s a graphical view of what’s going on:

The red squares indicate the stopping-point values that we calculated: the leftmost square (in the bottom left) is the value at the first stopping point, and the values progress to the right. It’s easy to see that they seem to be quickly zooming in on a particular value (indicated by the horizontal blue line), somewhere around 1.62.

The math word for this is convergence — the red squares (the values of successive stopping-points along the way to [1,1,1,1,\dots]) converge to a particular value. Technically, this means that we can get as close as we want to that particular value, as long as we are willing to pick a stopping point that is far enough along. The stopping point values will keep getting closer and closer (converging) to this particular value forever, even though they will never actually reach it exactly.

Note also that this is why the “value” of 1 + 1 + 1 + 1 + \dots is “infinity”. If we look at the value at successive stopping points (i.e. 1, 1+1, 1+1+1, etc.), they are not getting closer and closer to anything; they are simply getting bigger and bigger:

Stopping point values for 1+1+1+1+...

Saying that the value of something is “infinity” is really just a shorthand way of saying that it does not converge to anything (it diverges).

So when we ask for the “value” of the infinite expression [1,1,1,1,\dots], we are really asking: what do the stopping point values converge to?

Well, what do they converge to? You already know the answer to this: they converge to \frac{1 + \sqrt{5}}{2} \approx 1.61803, otherwise known as \phi!

There is much, much more going on here as well… but that will have to wait for another post. As always, feel free to comment with questions, ideas, comments, or anything at all (except spam).

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3 Responses to Convergence

  1. Elissa says:

    wheeeeeeeee!!!! This whole website came in perfect timing with me finally admitting my math dorkiness. Sweet!!! :-)

  2. Brent says:

    awesome. (=

  3. Loweeel says:

    Brent —

    I’ve just added your blog to my Bloglines and bookmarked it. I’ll be reading it as often as you update it, because it’s fantastic.

    – Lowell

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