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By Root 778 0

and Suiseth’s sequence, all the terms get closer and closer to zero. However, when Oresme tried to sum the terms in the sequence, he realized that the sums got larger and larger and larger. Even though the individual terms go to zero, the sum goes off to infinity. Oresme showed this by clumping the terms together: ½ + (1/3 + ¼) + (1/5 + 1/6 + 1/7 + 1/8) +…. The first group clearly equals ½; the second group is greater than (¼ + ¼), or ½. The third group is greater than (1/8 + 1/8 + 1/8 + 1/8), or ½. And so forth. You keep adding ½ after ½ after ½, and the sum gets bigger and bigger, and off to infinity. Even though the terms themselves go to zero, they don’t approach zero fast enough. An infinite sum of numbers can be infinite, even if the numbers themselves approach zero. Yet this isn’t the strangest aspect of infinite sums. Zero itself is not immune to the bizarre nature of infinity.

Consider the following series: 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 –…. It’s not so hard to show that this series sums to zero. After all,

(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) +…

is the same thing as

0 + 0 + 0 + 0 + 0 + 0 +…

which clearly sums to zero. But beware! Group the series in a different way:

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) +…

is the same thing as

1 + 0 + 0 + 0 + 0 + 0 +…

which clearly sums to one. The same infinite sum of zeros can equal 0 and 1 at the same time. An Italian priest, Father Guido Grandi, even used this series to prove that God could create the universe (1) out of nothing (0). In fact, the sequence can be set to equal anything at all. To make the sum equal 5, start with 5s and -5s instead of 1s and -1s, and we can show that 0 + 0 + 0 + 0 +…equals 5.

Adding infinite things to each other can yield bizarre and contradictory results. Sometimes, when the terms go to zero, the sum is finite, a nice, normal number like 2 or 53. Other times the sum goes off to infinity. And an infinite sum of zeros can equal anything at all—and everything at the same time. Something very bizarre was going on; nobody knew quite how to handle the infinite.

Luckily the physical world made a little more sense than the mathematical one. Adding infinite things to each other seems to work out most of the time, so long as you are dealing with something in real life, like finding the volume of a barrel of wine. And 1612 was a banner year for wine.

Johannes Kepler—the man who figured out that planets move in ellipses—spent that year gazing into wine barrels, since he realized that the methods that vintners and coopers used to estimate the size of barrels were extremely crude. To help the wine merchants out, Kepler chopped up the barrels—in his mind—into an infinite number of infinitely tiny pieces, and then added them back together again to yield their volumes. This may seem a backward way of going about measuring barrels, but it was a brilliant idea.

To make the problem a bit simpler, let us consider a two-dimensional object rather than a three-dimensional one—a triangle. The triangle in Figure 23 has a height of 8 and a base of 8; since the area of a triangle is half the base times the height, the area is 32.

Now imagine trying to estimate the size of the triangle by inscribing little rectangles inside the triangle. For a first try, we get an area of 16, quite short of the actual value of 32. The second try is a bit better; with three rectangles, we get a value of 24. Closer, but still not there yet. The third try gives us 28—closer still. As you can see, making smaller and smaller rectangles—whose widths, denoted by the symbol ?x, go to zero—makes the value closer and closer to 32, the true value for the area of the triangle. (The sum of these rectangles is equal to ?f(x)?x where the Greek ? represents the sum over an appropriate range and f(x) is the equation of the curve that the rectangles strike. In modern notation, as ?x goes to zero, we replace the ? with a new symbol, , and ?x with dx, turning the equation into f(x) dx, which is the integral.)

Figure 23: Estimating the area of a triangle