Terms of the series
We emphasize, again, two different kinds of mathematics.
Our tutor recently had a student dealing with infinite series. These can be puzzling to contemplate, so we’ll give a basic example. Suppose you add up a series of terms starting with 1 and halving each time: 1 + 1/2 + 1/4+ 1/8 + 1/16 + . . . You can see, we hope, that at each step you’re adding half the remaining distance to 2. Loosely speaking, you would have to add up an infinite number of terms to actually reach 2. More rigorously, mathematicians say that for any small distance from 2, we can add up a finite number of terms and get closer than that distance, so the limit of the series as the number of terms grows beyond any bound is 2.
(Classical Greek philosophers were wary of infinity, and would not have accepted that adding up an infinite number of numbers could result in something so mundane as 2. Infinity is tricky, and was only rigorously handled in the nineteenth century.)
Infinite series are widely used in the higher levels of mathematics. Indeed, there are many important numbers and functions defined by infinite series. And for most of them the limit is not as easy to work out as it is in our example; indeed, it may not be known at all. Mathematicians have been trying to find the exact infinite sum of inverse cubes, 1 + 1/8 + 1/27 + 1/64 + 1/125 + . . ., for centuries, without success.
But one doesn’t always need to know the exact value of the limit of an infinite series. When you ask your calculator for the sine of 36 degrees, it actually adds up a finite number of terms from the infinite series for sine, and displays the result. Most functions on your calculator work this way. For it to be useful you have to know how many terms to add up to get a result of the required accuracy. Given the immense practical importance of functions calculated through series, a lot of work has been done on this question. There are formulas to tell you that, given how many terms you’ve added up, the difference between your sum and the actual limit will be smaller than a certain amount. Our student was tasked with figuring out what this “error bound” was.
Her approach was direct: she would calculate the limit, and then subtract the eighteen-term (as it was) sum. Her question to our tutor was: how do I calculate the limit?
This reveals a more fundamental misunderstanding that she was aware of. She assumed that the exact answer could always be calculated. This impression is produced among students by having been given almost entirely problems where this is true, all the way from arithmetic through algebra to calculus. And when recourse must be had to a calculator, up pops the answer, reinforcing the impression. No doubt the many labs in Chemistry and Physics that call for a “percent error” by reference to the “true value” also contribute.
This misunderstanding confuses two kinds of mathematics, which we’ve called the Greek and Babylonian. The first gives exact, provable results, though not always practically useful ones; the second, approximations good enough for the job at hand. Combining the approaches (as in calculating the error bound) is very powerful. Confusing them is, we think, dangerous. Even the best approximations are not exact, and you need to know what your uncertainties are.
