Halving series

Note: this article makes use of MathML, the standard XML markup for math formulas. Sadly, this is not properly supported on some allegedly ‘modern’ and ‘feature-rich’ browsers. If the formulas don't make sense in your browser, consider reporting the issue to the respective developers and/or switching to a standard-compliant browser.

This article has been inspired by a post on /r/math, that presented a graphical way to show that the sum of the series of consecutive powers of 1/2 (geometric series with ratio 1/2) converges to 1:

\sum_{n = 1}^{∞} \frac{1}{2^{n}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1

The proposed image was very roughly made and raster, so I decided to re-create it, cleaned up and in vector form:

A square subdivided into triangles, each taking half of the space left by the others — Halving series (triangles)

Of course, this is not the only possible approach to demonstrate (note: demonstrate, not prove) the convergence of the series. A similar approach (the one I was more familiar with) is based on rectangles rather than triangles:

A square subdivided into rectangles, orthogonally to each other, each taking half of the space left by the others — Halving series (rectangles)

Which one of the two is ‘better’ is obviously up to debate: some people prefer the triangle version, because the series is assembled from figures which are (geometrically) similar, while others prefer the rectangle version, as it features lines parallel to the sides of the original square, and thus clearer features to a higher detail (i.e. more visible sections).

Of course, these are not the only possible representations for the series; for example, sticking to rectangles, one can also proceed by halving each remaining fraction of the original square with lines all parallel, obtaining the striped version:

A square subdivided into rectangles, parallel to each other, each taking half of the space left by the others — Halving series (stripes)

I honestly find this version boring compared to the other two, since in this case it makes much more sense to give a one dimensional representation, by consecutive halving of segments, instead:

Two segments of length 1. The first is divided into smaller segments, each half of the preceding one — Halving series (segment)

The segment splitting is most definitely the simplest, but some people might be more comfortable with the two-dimensional demonstrations, since apparently humans tend to deal better (intuitively) with areas than with lengths.

Animations

Note: this section stress-tests support for SVG animation with SMIL. Sadly, this is not properly supported on some allegedly ‘modern’ and ‘feature-rich’ browsers. If the animations don't make sense in your browser, consider reporting the issue to the respective developers and/or switching to a standard-compliant browser.

One of the upsides of using SVG as format for these demonstrations is that it is possible to animate them. Animations have two advantages in cases such as these: one, they present a more intuitive visualization of the “progress” of the infinite summation; and two, they make it easier to avoid the mess which is “perfectly centered text” in SVG (an issue that deserves its own rant).

We can animate the one-dimensional example by introducing each new term with a timing which is proportional to the length of the segment, obtaining something like this:

Animated version of the halving series represented with segments — Animated halving series (segment)

There's no need for textual explanation, although we can achieve it by adding an animated summation (which sadly doesn't seem to work correctly in most browsers):

Animated version of the halving series represented with segments, with numbers on each new segment — Animated halving series (segment, with formula)

Of course, the same can be done in two dimensions, with both the triangles and rectangles approach.

Animated version of the halving series represented with triangles — Animated halving series (triangles)

Animated version of the halving series represented with rectangles — Animated halving series (rectangles)

(And of course we could have the variants with formulas here as well, but I think the idea is clear and, honestly, coding these SVG and MathML by hand gets boring after a while, so their realization is left as an exercise to the reader.)