When we add a sequence of numbers together, we typically expect to get the same sum, regardless of the order in which these numbers are added.
But, there are certain types of infinite series, where by changing the order in which the numbers are added, we can change the sum to any number we wish.
To understand this, we need to understand the difference between absolute convergence and conditional convergence.
Suppose we have a square with an area of one.
This square can be thought of as being composed of an infinite number of rectangles.
The sum of the areas of all these rectangles must be equal to the sum of the original square.
Therefore, in this example, we have an infinite sequence of numbers that adds up to one.
Let’s now change this by multiplying every other number in the sequence by negative one.
We now get a new sum.
In this example, we say that this series converges absolutely, because when all the values were positive, the series still converged to a finite value.
When a series converges absolutely, we will always get the same sum regardless of in what order we add the numbers in the sequence.
This is true for both the original series where all the terms were positive, and the new series in which half the terms are negative.
Let’s now consider a completely new example.
Here, we have a new sequence of numbers.
The sum of this series does not converge to any finite number.
The sum of this series approaches positive infinity.
Let’s now multiply every other number in this sequence by negative one.
The sum of this series now converges to a finite number, which happens to be equal to the natural logarithm of two.
In general, when we have a sequence with a sum that approaches infinity, then we might or we might not get a finite sum when we multiply every other number in the sequence by negative one.
In this example, we get a finite sum, and we therefore say that this sequence converges.
However, this sequence does not converge absolutely, because it did not converge when all the terms in the sequence were positive.
We call this conditional convergence.
All sequences with conditional convergence have a very interesting property.
By rearranging the order in which we add the numbers in the sequence, we can change the sum.
By rearranging the order in which we add the numbers in a conditionally convergent sequence, we can make the sum to be equal to any real number that we want.
By rearranging the order in which we add the numbers in a conditionally convergent sequence, we can also make the sum equal to negative infinity.
By rearranging the order in which we add the numbers in a conditionally convergent sequence, we can also make the sum equal to positive infinity.
This is possible because our sequence of numbers actually consists of two different sequences.
One of these sequences consists exclusively of positive numbers, that add up to positive infinity.
One of these sequences consists exclusively of negative numbers, that add up to negative infinity.
If it happens to be possible to arrange these numbers in a way that results in a finite sum, then it is also possible to rearrange these numbers so that we get any sum we desire.
And it is also possible to rearrange these numbers so that the sum approaches positive or negative infinity.
The fact that infinite sequences can have such bizarre behavior has led some philosophers to conclude that infinity is a concept that can exist only in mathematics, and not in the actual physical world.
Though, many others disagree.
Whether or not the Universe can be infinite is discussed in the video "Infinite Universe?"
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