What’s a Convergent Series?

“Convergent Series” is a term from mathematics that, in my opinion, is wonderfully appropriate to metalsmithing and, in particular, to working with metal clay.

Quick Math Lesson

For years I taught a college-level math course that included an introduction to “infinite series” which include both “convergent” and “divergent” kinds of series. it was one of the most challenging topics covered in that year’s material, but I always tried to make it a much more manageable topic through the use of both narrative and various visualization tools. I don’t have access to the latter right at the moment (I’ll keep trying and can update this later on), but here’s a bit of the verbal part.

First of all, we’d consider sequences. That’s just a string of individual numbers that follow some pattern, such as:

• The counting numbers (aka positive integer):
1, 2, 3, 4, 5, etc., or
• Square numbers (1*1, 2*2, 3*3, 4*4, …):
1, 4, 9, 16, 25, etc., or
• Alternating square numbers (positive, negative, positive, negative, …)
-1, 4, -9, 16, -25, etc., or
• Reciprocal counting numbers (fractions with 1 on top and the counting numbers on the bottom):
1/1, 1/2, 1/3, 1/4, 1/5, etc., or
• Alternating reciprocal squares:
1/1, -1/4, 1/9, -1/16, 1/25, etc, and so on

Now, sequences were fairly simple, a rather easy way to make sure everyone understood the terminology, but then we’d move onto series, which are what you get when you add together increasingly longer groups of the elements in a sequence. For example;

• 1 + 2+ 3 + 4, 5+ …, or
• -1 + 4 + (-9) + 16 + (-25) …, or
• 1/1 +( -1/2) + 1/3, +(-1/4) +1/5 + (-1/6)…, or
• 1/1, + (-1/4) + 1/9 + (-1/16) + 1/25 …, and so on.

And the question was: if you kept adding on more and more terms, forever, was there any way to know what the total would be? We would then spend a month or more going over lots of different ways to figure out the answer to that (in part because how you do so depends on tricky things about the series you start with).

Occasionally, the final answer would be “indeterminate” which meant that there really was no way to know what a series would do. Sometimes, we’d discover that the series (or its absolute value…) would just keep getting bigger and bigger and bigger forever: that’s a divergent series.

But the really fun part (fun in the math sense, that is) was when we could find that a series, even had we kept adding to it forever, would eventually reach some limit, some state that it wouldn’t go past, couldn’t go past, some actual value that we could determine. In that case, we had a convergent series, a series that would converge to that limit.

How Does That Apply to This Art?

From the very first time I ever got my hands on metal clay, started to play with it, and came to understand what it did, I started thinking of it as a “convergent series”!

From CS’s Learning Metal Clay (shinier than this photo would indicate)

You work with it for a while, you develop your design, you add a bit here and there (and, at times, you add a bit, take a bit away, add a bit more, take another bit away…), do some of this and some of that, heat it for a while and then a while longer, let it cool and tweak it a bit more, and so on. While it’s not truly an infinite set of steps — sometimes it is tempting to keep reworking a piece but rarely does that really improve it — in theory you could keep going like that forever. But whether you work it for a finite time or a seemingly infinite one, there would still be a limit: no longer would you have any clay at all and the piece you’d been working on would “converge” to fine silver. It reaches that state but (as long as you haven’t broken the rules of the series, e.g., have not gotten it so hot that it has changed out of its solid state…) it can’t go beyond that. Which is great: Your end result is a piece of wonderful, pure, fine silver.

Finishing such an item is just as “magical” as the convergence that mathematicians work with but, really, it’s way, way more fun!

(Plus, speaking of amazing things, “Convergent Series” just happens to have the same initials as does my name, “Carol Scheftic,” so I can sign this note with those letters and it works both ways!)

-cs