We know Infinities are very tricky things and also have perplexed mathematicians and philosophers for thousands of years. Sometimes the sum of a never ending list of numbers will become infinitely large and sometimes it will get closer and closer to a definite number. And the other hand it will defy having any type of definite sum at all. A little while ago my friend was giving a talk about infinity that included a look at the simple geometric series below :
Square = ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + … and so on ….
Forever, every term in the sum exactly half the size of it’s predecessor. The sum of this kind of series is actually equal to 1, but someone in the audience who was not a mathematician wanted to know if there was any way to show him that this sum is true.
Fortunately there is a simple demonstration that just uses a picture. Just draw a square of size 1 x 1, so its area is 1. Now let’s divide the square in half by dividing it from top to bottom into two rectangles. Each of them must have an equal area to ½ . Now divide one of these rectangles in two to make two smaller rectangles, each with area equal to ¼ .Now divide one of these smaller rectangles in half again to make two more rectangles, again each of area equal to 1/8. Keep on going like this, making a rectangle of half the area of the previous one, and look at the picture. The original square has just had its whole area sub-divided into a never ending sequence of regions that fill it completely. The total area of the square is equal to the sum of the areas of the pieces that my friend have left intact at each stage of the cutting process, and the areas of these pieces is just equal to our formula “Square” – above. So the sum of the formula “Square” must be equal to ONE (1) –> the total area of the square.
In-fact when we encounter a formula like “Square” for the first time, we work out its sum in another way then we notice that each successive term is one half of the previous one and then multiply the whole formula by ½ so we have :
½ x Square = ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + … and so on ….
But we notice that the formula on the right is just the original formula, Square, minus the first term which is ½ . Finally we have that ½ x Square = Square – ½ , and Square = 1 again.
Is it easy ? think again.