In mathematics, we understand a series as the sum (which can sometimes be infinite) of terms in a sequence. We can say that a series is an ordered set of elements of this sequence combined by the addition operator. The term “infinite series” is used to emphasize the fact that the series contains an infinite number of terms.
The symbol Σ (summation) is used to denote a sum of N terms of a sequence. After the Σ symbol, the lower index of the first term to be added appears and then the upper index of the last term to be added.
For example, be the sequence called APPLE with elements 2; 4; 6; 8; 10; 12; 14; 16; 18; 20; 22; 24… The first term of this sequence is 2, the second term is 4, the tenth term is 20. So when we write Σi=2i=10 APPLE, it means adding from the second term of the sequence to the tenth term of this sequence:
And here comes a joke by mathematicians … Once a mathematician went to the televisions factory and added the registration numbers for each device that passed in front of him. After an hour he said he was ready and that he could sell his creation. Did they ask what he created? And he replied, that he had just produced his “TV series”.
But series is one of the cool and fun things in mathematics (different from my jokes). The series supports many tools for working with them. One of them is the super power of “telescoping” series!
Suppose I have a series Σi=1 BEET, (when we do not define a maximum value in the series, this means that we consider any number to be its maximum). To remember, a series written with this rule, should look like this:
b1+b2+b3+b4+b5+…+bN, where N is any number.
To show how “telescoping a series” works, let’s say that bi is ai+1 – ai (remembering that the APPLE sequence is the increasing even numbers starting from 2). Thus, we can rewrite the series Σi=1 BEET as:
(a2 – a1)+(a3 – a2)+(a4 – a3)+(a5 – a4)+(a6 – a5)+…+(aN – aN-1)+(aN+1 – aN).
But this sum, by the commutative property of addition, can be rewritten as follows:
-a1+a2 – a2+a3 – a3+a4 – a4+a5 – a5+a6 – … – aN-2+aN-1 – aN-1+aN – aN+aN+1.
To facilitate the visualization, I will put the groups that will be canceled in parentheses.
-a1+(a2 – a2)+(a3 – a3)+…(aN-2 – aN-2) +(aN-1 – aN-1) +(aN – aN) + aN+1.
As we can see, all terms, except the first and the last, will be canceled. That is, when we telescope the series Σi=1 BEET that we defined, we have:
-a1 + aN+1.
For example, if our final N is 100, we have that the sum of these differences is:
-a1 + a101 = -2 + 202 = 200.
But you may be suspicious of this super power of “telescoping” series, after all, in this example, all bi (for i any number) will be equal to 2 (since it is the difference between two terms in the pair sequence).
Let’s take a more “chaotic” sequence, how about prime numbers (that is, Natural numbers greater than 1 that are divisible only by itself and by 1)? We will call this sequence P with the elements p1=2; p2=3; p3=5; p4=7; p5=11; p6=13; p7=17; p8=19; p9=23; p10=29; … Thus:
bN = pN+1 – pN.
So, when we take the series Σi=1i=9 BEET, for example, we have:
p2 – p1+p3 – p2+p4 – p3+p5 – p4+p6 – p5+p7 – p6+p8 – p7+p9 – p8+p10 – p9.
Using the commutative property of addition, we rearranged our series as follows:
-p1+(p2 – p2)+(p3 – p3)+(p4 – p4)+…+(p7 – p7)+(p8 – p8)+(p9 –p9)+p10
Thus, from this series with less behaved bi values, we have the following result:
-p1 + p10 = -2 + 29 = 27
Similarly, if our series were Σi=2i=8 BEET, we would have the solution by “telescopy”:
-p2 + p9 = -3 + 27 = 24
The idea here is that there are series that resemble a retractable telescope, that after closing the instrument, we have only the beginning and the end. Similar to storing a retractable telescope, the other parts retract into it.
Ok… this is cool, but what’s the use?
When the series is finite, this can be replaced by the exhaustive sum of all its terms … boring, but it works. But when the series is infinite, we need this technique to solve it analytically (we cannot simply add infinitely). A cute example of an infinite series is.
Σi=1i=∞ CABBAGE, with its elements expressed by:
Cn = 1/n(n+1).
The first terms of this sequence are:
1/2; 1/6; 1/12; 1/20; 1/30…
The sum of these 5 terms above is 0,83…
The sum of these 10 terms is 0,9090…
The sum of these 100 terms is 0,99009900
Will it arrive to 1? A good guess would say yes, but a “telescope” in this series PROVES THAT YES. The idea to telescope a series see it as a subtraction within the terms of a sequence. For example:
Cn = 1/n(n+1) = (1/n) – (1/(n+1)).
So, the series Σi=1i=∞ CABBAGE, looks like this:
1/1 – 1/2 + 1/2 – 1/3 +1/3 – 1/4 + 1/4 – 1/5 + … – 1/N + 1/N – 1/(N+1).
Through the “telescopy” of this series, it is reduced to:
1/1 – 1/(N+1).
And when N is ∞, 1/(∞+1) will be very close to 0. So this series when N goes to infinity, results in:
1/1 – 0 = 1.
The next figure is for a reflection to the end of this text, with reference to the artist René Magritte who in 1929 made a work on the duality of object and representation. Illustrating a pipe and writing next to the image “Ceci n’est pas une pipe”, which can be translated as “This is not a pipe”.
This generated a lot of repercussions at the time, because in the picture a pipe was clear and obvious, so how could that not be a pipe? The author’s response was simple and sufficient to justify his claim to his entire audience. If this is a pipe, then let someone try to smoke it. The rhetoric that the painting itself is a representation of a pipe was the message desired by the work. Other works and references to this work appear in several areas, including this work in which I present a representation of a telescope like the telescopic series. That despite her “apparently” retracting, this is just a representation that is associated with the form of retraction. In fact, the series does not physically retract like a telescope or other object, it is itself an abstract object and its retraction is also done in the same way, surpassing any analogy to the concrete universe.