If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
Niels H. Abel
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Yesterday, we had a birthday party and celebrated with the following cake. Avirupa took half of it and Ravi half of the remaining half. Then the turn for Ritesh, Chandrani and others (in the same way as discussed before). Of course, so many guests were there. Do you think, was it possible to have an amount of cake to everyone?
Let’s observe this situation from a mathematical point of view. When Avirupa took half (or, 1/2) of it, then half of the cake remain.
Since Ravi took half of the remaining half (or, 1/2 of 1/2, i.e., 1/4), a quarter (or, 1/4) of the cake is still left out.
Dividing the left over cake again in the same way will give us the following diagrammatic representation. Note that, with each new piece they take, in this way, the amount of cake left over is halved.
Processes which go on for ever are of frequent occurrence in mathematics and without them, the subject would be very different. Below, we will show the amount of cake taken after first, second, third and fourth takeout.
Finally note that, dividing up the cake by first taking half the cake, then a quarter, then an eighth, and so on, gives us the sum
This is the common way to represent an infinite series using the summation notation and the dots indicate that we need to divide the cake into halves, in the obvious way, without ever stopping. At first sight, perhaps, this SUM is bound to be infinite, because all the individual terms are positive and we are adding up an infinite number of them.
We can see, while adding larger and larger number of terms, the partial sums of this series is approaching towards a value (= 1 here, the whole birthday cake). This is a good example of infinite geometric series. Recall that the sum of an infinite geometric series with first term a and common ratio r is given by
Then we can find the value of S as shown below.
Of course, the above series has an infinite number of terms, but we can comprehend how small each of them is. The figure below shows pretty well what’s happening. We can see exactly how each corresponding term takes up a smaller and smaller area, somehow making it converge to an area of exactly 1.
Suddenly, then, two things seem clear. First, we are never going to get the whole cake by this procedure. Second, we can, however, get as much of it as we like by taking enough pieces. And this is, essentially, exactly what mathematicians mean when they say that converges to the value 1.
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