By Tom Brown
The essay is meant for people with some after-school mathematics training and just to understand some theory it is not meant as rigorous at all just to help grasp some basic ideas. We rely mostly on intuition. These insights are invaluable for university coursework also I plan some more essays where this work would be very helpful with some surprising results as in astrophysics.
The method of mathematical induction is an indispensable tool for proving results, in this case formulas.
The infinite geometric series 1 + r + r ² + r ³ + … is 1 + ½ + ¼ + … = 2 for r = ½ and for r is convergent when -1 < r < 1 and the sum then is 1/(1 – r).
It is easy to visualize. We begin with a unit square, that is sides of unit length and area of 1² = 1. By dividing it into two equal rectangles and colouring the one we have a shaded area of ½.
Now dividing one of these rectangles into two squares and shading gives ½ + ¼ and once more is 1/2 + 1/4 + 1/8. Subsequently the shaded area is 1/2 + 1/4 + 1/8 + 1/16 … if you have made a figure it is clear that the shaded (filled) part eventually fills the whole original square. The infinite sum of areas therefore is finite and is one unit squared. A simple sketch will make it clear.
This type of series occurs frequently in everyday life. Unfortunately there it often is not bounded this depending then on whether you are a bank or a borrower! Think in terms of compound interest.
For the well known arithmetic series the sum of n terms is 1 + 2 + … + n = ½n(n+1). This kind of series does not converge the sum is unbounded. Typically we might use mathematical induction to prove that a given partial sums expression is correct and if so may then calculate the infinite sum.
As another example for which we will have use elsewhere also the question is whether (a “harmonic” series) the infinite sum of the squares 1/r² is convergent. And therefore adds up to a finite and fixed number. Convergence itself is easy to show with the integral test. The series is not so easy to calculate but the value is
1 + 1/4 + 1/9 + 1/16 + ... + 1/r² + ... = pi²/6
By making r larger and larger the sum approaches a limit and to any desired accuracy. We will find use of this series in another submission. You can calculate the partial sums and estimate the limit with just a calculator and plausibly verify convergence. To find the exact total one must thus formally add up this series which you can, since it is convergent and must be a finite number.
Once we do have the formula for the first n terms verifying it and then evaluation of such an infinite sum still appears very complicated. Yet often it is just a matter of algebraic manipulation and then is very easily confirmed. A formula may routinely be proved as correct with mathematical induction and the series established as convergent. In a way it is necessary for you know the answer before the question!
There are advanced tools such as Laplace transforms and in complex variables and also the very elegant method of generating functions in discrete mathematics to calculate series. Many sums can even be evaluated with just high school algebra but it can get very tricky.
Please note it is not necessary to actually calculate the numerical value of a series we can know the series converges without necessarily knowing the actual finite number. Indeed there are simple tests in a standard first-year syllabus which can determine most types of series as convergent or not, or diverges to ± infinity.
A series that is increasing and bounded from above is necessarily convergent. Again a sketch will help. This is a formulation of the completeness axiom for real numbers.
Also computer evaluation of functions are made by power series this is how a machine calculates logarithmic functions and trigonometric and hyperbolic and roots numerically the function values are not all in some kind of infinite bank. All a computer can really do is addition, subtraction and then multiplication and division.
There are many more examples and in almost every branch of mathematics. These ideas are necessary intuition for understanding and applying calculus and elementary algebra.