Infinite series are a mind-boggling aspect of mathematics that explore a world of unconventional manipulations and results. While mathematicians have found ways to agree on the sums of some common infinte series, there are many more that continute to either elude mathematicians or give amusing results. Take for instance, the infinite series above that follows a valid mathematical proof but ends up showing 2 = 1. So, what is this weird sequence and can we find any more of these?
Common Infinite Series
Some infinite series are as straightforward as infinity can get. Let's start with the most famous one:
Achilles is a sprinter. He needs to reach a red flag. First, Achilles manages to cover half the distance. Next, he covers half of the remaining distance (1/4 of the total). Then, he covers a further half of the remaining (1/8 of the total). This goes on and on, as Achilles keeps covering the infinite half subdivisions of the track (1/16, 1/32, 1/64...) and thus logically never reaches the end. Achilles argued there are infinite subdivisons hence the race never ends.
This situation can be set up as follows:
This sum is famously explained using this diagram. This came from a realisation that if a sum keeps getting closer and closer to a certain value as more and more terms are added, then that value is called the limit of the sum.
Notice how all the small rectangles add up to a neat square of side length 1. Hence, we say that this infinite series sums to 1 since each additional rectangle makes the sum approach 1. This means the series converges to a value.
This concept is commonly taught in high school syllabi hence we take this discussion one step further.
Alternating Series
An alternating series is one where each term alternates between signs (+ and -). The most famous example of an alternating series is the alternating harmonic series:
Out of the scope of this post, this series has been shown to converge to a value. For those interested, do explore Leibiniz's alternating series test. Interestingly, however, if we eliminate the alternating nature of the series, it turns out to be divergent, not convergent.
This is an interesting realisation to note, the fact that an alternating series is convergent, but its absolute value counterpart is not. This is known as conditional convergence.
Rearrangement Theorem
Riemann conjectured and went on to prove that the sum of such a conditionally convergent series depended on the order of its terms. This further meant that such a series could be shown to sum to any real number.
Imagine! The same sum, the same set of numbers all approaching one single value... but that value can be anything you want. That is the beauty of the Riemann's Rearrangement Theorem. Let us take the same example we already explored. Set the value of this sum to be "S".
Now, we can do some algebraic manipulation to the same alternating series. The first step changes the order of the terms (as indicated by the red and green for your ease). Next, we subtract adjacent terms (as indicated by the brackets). Finally, we take a factor of 1/2 common. This allows us to show the same series that sums to S in one order of terms, summing to half of S in another rearrangment.
For the curious minds, the value of S has been shown to be ln(2). While this is one algebraic manipulation, Riemann proved that such series could in fact be arranged to sum to ANY real number. It is important to now discuss what this means.
The mathematical community has not painstakingly shown a rearrangement of this series for every real number, because there are infinite (and that is a huge amount). Instead, Riemann had a very straightforward proof worth understanding.
Proving the Rearrangement Theorem
This is a very simplified version of the proof since it is made to be more accessible. Hence, I also avoid most rigorous mathematical notation and do not generalise it instead focusing on the harmonic series example above.
Let us start by taking the conditionally convergent series and segregating it into positive and negative terms. This gives us two separate series, P being all positive terms and N being all negative terms.
Note that this means S is the sum of both P and N (S = P + N). Now, using some complex logic outside our scope, Riemann showed both P and N were not convergent. This meant both these series do not approach a finite value. This is not very important for most hobbyists to grasp.
So, let us now get to the magic. Pick any number, for example we pick 1.2. Remember, the series of positive numbers (P)? We now start adding its first few terms. We keep adding its terms UNTIL we first go above 1.2 in value. Let's use the harmonic series to understand what is being said.
After adding two terms, we crossed the 1.2 treshold hence we stop adding more terms. Now, we add the negative numbers (N) UNTIL we first go below 1.2 in value. And so on and so forth.
As you will notice, the sum is becoming closer and closer to our desired value of 1.2. This is an infinite process and can keep going on forever until, theoretically, we reach the exact value of 1.2 for the sum. Riemann had just shown how a conditionally convergent series could theoretically converge to any value just by rearranging the positive and negative terms to reach the desired real number eventually!
Conclusion
While infinite series in mathematics bring with them a lot of oddities irrespective, this unique property is one of particular amusement. This property of infinite rearrangement to produce infinitely many values from the same series adds another layer of ambiguity and mystery to the fascinating world of infinity.
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