Geometric series: means to no end
You might have heard this old nursery rhyme:
As I was going to St. Ives,
I met a man with seven wives,
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits:
Kits, cats, sacks, and wives,
How many were there going to St. Ives?
I hope the wives also had many husbands and that the cats were comfortable. Not including the narrator, we can calculate the total as
This sum of increasing powers is called a geometric series. A geometric series is the sum of a geometric sequence, defined as[1]:
n is the number of terms in the sequence, which includes the first term which is the zero-th power, 1. So the number of things going to St Ives is
We could add all those terms. But maybe there is a shortcut like there was for arithmetic series. If you want to find the shortcut on your own, have a go before reading on. Find a hint after the cat.
Hint: look for the recursive structure in the sum.
A solution: aside from the recursive definition of the series:
there is another recursive relation:
which we can massage as follows:
Et voila. We have a nice formula for a geometric series. So our answer to the St Ives problem is this, right?
Wrong! The riddle never says that all the mentioned things are going to St Ives, only the narrator is going for sure ;)
But why does there happen to be a neat formula for a geometric series? It seems too convenient to be true. I think it’s because the definition gives rise to a property or structure that is not a mere restatement of the definition. The structure inherent in a geometric series is
This is a self-similar structure, like a fractal, which is why we see geometric series when we calculate, for example, the area of fractals.
The formula for the area of the Koch snowflake, above, is as follows, where 1 is the area of the original blue triangle:
because the 3 green triangles have 1/9 the area of the blue triangle, the 12 yellow triangles have 1/9 the area of the green triangles etc. We can rewrite the formula like this so that the geometric series becomes obvious
Although the series is infinite, the total area is not infinite because the terms keep getting smaller. The area is thus
This works out because the term 4/9^∞ is practically zero. (Or “in the limit approaches zero”.) In fact, the sum of an infinite a geometric series with r <1 is:
Pretty useful. Now we can answer things like Zeno’s paradoxes. The Dichotomy Paradox says you can’t get anywhere because to travel any distance you have to have traveled half that distance, and half that distance, ad infinitum.
The sum of the steps, and the corresponding units of time needed to take those steps, is an infinite sum with a finite limit. We can calculate the total time and distance as:
We traverse the infinite every day…
But why are they called Geometric Series? What is particularly geometric about them?
Perhaps it has to do with the increasing dimensions, but the dimensions get pretty high pretty quickly and high-dimensional geometry is some pretty abstruse stuff. This thread led me to some interesting possible answers. ‘Arithmetic’ and ‘geometric’ series seem to relate to arithmetic and geometric means. The arithmetic mean is the sum of numbers divided by the count. The geometric mean is the product of the numbers divided by the count. For the simple case of two numbers:
The idea of the geometric mean goes back to Euclid. For two numbers, it can be thought of as “squaring a rectangle”: finding the side of a square whose area is the same as the rectangle of the 2 numbers you want to find the mean of. In a right-angle triangle, the geometric mean of the segments p and q is the height h:
There’s a neat proof that you can find the geometric mean, c, of a and b, by drawing the circle where a+b is the diameter like so [2]:
The geometric mean is quite useful, like for comparing things that have multiple dimensions you want to compare. It’s like comparing different shapes using their area.
In an arithmetic sequence each number is the arithmetic mean of the numbers either side. In a geometric sequence each number is the geometric mean of the numbers either side. So they are a series of means! The use of the term ‘geometric progression’ goes back to Michael Stifel’s book “Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis” in 1543, or one with a cooler title, “Whetstone of Witte” in 1557 by Robert Recorde (which also contains the first recorded use of the equals sign!).
So the name could be due to the geometric mean. It could also be because the geometric series has a lot of neat geometric depictions:
The above two pictures are particularly cool because they demonstrate pictorially that the following geometric series converges to 1/3 the size of the containing shape [3]:
Archimedes used this geometric series in the 3rd century BC to calculate the area enclosed by a parabola and a straight line. He approximates the area with an inscribed triangle, and at each generation generates more triangles whose areas are 1/8th of the original. The successive triangles form a convergent geometric series, like a fractal, so that the answer is 4/3 of the blue triangle.
Geometric series have been known in geometry for a long time. But the earliest recorded reference is in the Rhind Mathematical Papyrus, scribed by Ahmes around 1550BC. I love the opening sentence to this papyrus:
Correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.
Sign me up to that cult! Maybe there’s a discount for pythagorean cult members or if you can solve its problem 79:
An estate’s inventory consists of 7 houses, 49 cats, 343 mice, 2401 spelt plants (a type of wheat), and 16807 units of heqat (of whatever substance — a type of grain, suppose). List the items in the estates’ inventory as a table, and include their total.
“Ahmes performs a direct sum, but he also presents a simple multiplication to get the same answer: 2801 x 7 = 19607”. Exploiting the relation:
They didn’t have the concept of exponents, so it would have been hard for them to discover the hidden formula.
Next mathematical history mystery: what’s up with cats and the number 7?
[1] The generic formula actually contains a constant a which makes the first term a instead of 1, and the last term ar^{n-1} instead of r^{n-1}. It can be factored out, so I leave it out for simplicity.
[2] You can easily prove that the height of a right-angle triangle is the geometric mean of the segments in the base that it intersects, and you can easily prove (Thales theorem) that you can form a right-angle triangle from any point on the circle where the line segment is the diameter.
[3] The gray, white and black squares all depict the sequence, and they each make up a third of the square.
[4] I read that the Rhind papyrus contained a mention to geometric series via http://mste.illinois.edu/courses/ci499sp01/students/ambucher/math306geo.pdf
[5] The recursive formula an infinite geometric series doesn’t make sense if r > 1, just like 1/0 doesn’t make sense.
[6] I didn’t understand why the wikipedia article on geometric series mentions a proof of the formula using a made-up “telescoping” series. I’m guessing that it’s obvious that telescoping series converge, and it’s just another relation that shows geometric series converge.
[7] The wikipedia article on geometric series makes a nice point that decimal fractions are geometric series, and we can use the geometric series formula to have another proof that 0.99999… = 1.
[8] All this is only scratching the surface of the historical usefulness of the geometric series. I have to figure out how Wallis used them to find areas under hyperbolas, and how Newton used them to divide polynomials.
Why am I thinking about this? Because I wanted to prove the mortgage payment formula.
Thanks to everyone on the internet for assembling all these facts, and to Justin Kao for http://mathurl.com/.
