TELESCOPIC SUMMATION: HOW IT IS SOLVED AND EXERCISES SOLVED - MATHS - 2024

The sum telescopic is a branch operations numerical series. It deals with the summation of elements from an initial value to “n” of expressions whose argument obeys any of the following patterns:

(F _x - F _{x + 1}); (F _{x + 1} - F _x)

As also:

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They represent a summation of elements that when developed, is subjected to cancellations of opposite terms. Making it possible to define the following equality for telescopic summations:

Its name comes from the relationship with the appearance of a classic telescope, which could be folded and unfolded, notably changing its dimension. In the same way, telescopic summations, which are infinite in nature, can be summarized in the simplified expression:

F ₁ - F _{n + 1}

Demonstration

When developing the summation of terms, the elimination of factors is quite obvious. Where for each of the cases, opposite elements will appear in the next iteration.

The first case, (F _x - F _{x + 1}), will be taken as an example, since the process works in a homologous way for (F _{x + 1} –F _x).

Developing the first 3 values ​​{1, 2, 3} the trend of simplification is observed

X ₁ (F ₁ - F _{1 + 1}) = F ₁ - F ₂

X ₂ (F ₂ - F _{2 + 1}) = F ₂ - F ₃

X ₃ (F ₃ - F _{3 + 1}) = F ₃ - F ₄

Where when expressing the sum of the elements described:

X ₁ + X ₂ + X ₃ = F ₁ - F ₂ + F ₂ - F ₃ + F ₃ - F ₄

It is observed that the terms F ₂ and F ₃ are described together with their opposites, which makes their simplification inevitable. In the same way, it is observed that the terms F ₁ and F ₄ are maintained.

If the sum was made from x = 1 to x = 3, it means that the element F ₄ corresponds to the generic term F _{n + 1.}

Thus demonstrating equality:

How is it solved?

The purpose of the telescopic summations is to facilitate the work, so that it is not necessary to develop an infinite number of terms, or to simplify some chain of addends that is too long.

For its resolution it will only be necessary to evaluate the terms F ₁ and F _{n + 1}. These simple substitutions make up the final result of the summation.

The totality of the terms will not be expressed, becoming necessary only for the demonstration of the result, but not for the normal calculation process.

The important thing is to notice the convergence of the number series. Sometimes the summation argument will not be expressed telescopically. In these cases, the implementation of alternative factoring methods is very common.

The characteristic factorization method in telescopic additions is that of simple fractions. This occurs when an original fraction is decomposed into a sum of several fractions, where the telescopic pattern (F _x - F _{x + 1}) or (F _{x + 1} - F _x) can be observed.

Decomposition into simple fractions

To verify the convergence of numerical series, it is very common to transform rational expressions with the simple fraction method. The goal is to model the plot into the shape of a telescopic summation.

For example, the following equality represents a decomposition into simple fractions:

When developing the number series and applying the corresponding properties, the expression takes the following form:

Where the telescopic shape is appreciated (F _x - F _{x + 1}).

The procedure is quite intuitive and consists of finding the values ​​of the numerator that, without breaking the equality, allow us to separate the products found in the denominator. The equations that arise in the determination of these values, are raised according to comparisons between both sides of the equality.

This procedure is observed step by step in the development of exercise 2.

History

It is quite uncertain to be able to define the historical moment in which the telescopic summations were presented. However, its implementation begins to be seen in the seventeenth century, in the studies of numerical series carried out by Leibniz and Huygens.

Both mathematicians, exploring the summations of triangular numbers, begin to notice trends in the convergence of certain series of successive elements. But even more interesting is the beginning of the modeling of these expressions, in elements that do not necessarily follow one another.

In fact, the expression used previously to refer to simple fractions:

It was introduced by Huygens and immediately caught Leibniz's attention. Who over time could observe the convergence to the value 2. Without knowing it, he implemented the telescopic summation format.

Exercises

Exercise 1

Define to which term the following sum converges:

When manually developing the sum, the following pattern is observed:

(2 ³ - 2 ⁴) + (2 ⁴ - 2 ⁵) + (2 ⁵ - 2 ⁶)…. (2 ¹⁰ - 2 ¹¹)

Where the factors from 2 ⁴ to 2 ¹⁰ present positive and negative parts, making their cancellation evident. Then the only factors that will not be simplified will be the first “2 ³ ” and the last “2 ¹¹ ”.

In this way, when implementing the telescopic summation criterion, the following is obtained:

Exercise 2

Transform the argument into a telescopic type summation and define the convergence of the series:

As indicated in the statement, the first thing to do is to decompose into simple fractions, in order to restate the argument and express it in a telescopic way.

You must find 2 fractions whose denominators are respectively "n" and "n + 1", where the method used below must obtain the values ​​of the numerator that satisfy the equality.

We proceed to define the values ​​of A and B. First, add the fractions.

Then the denominators are simplified and a linear equation is established.

In the next step, the expression on the right is operated, until a pattern comparable to the “3” on the left is achieved.

To define the equations to be used, the results of both sides of the equality must be compared. In other words, no values ​​of the variable n are observed on the left side, in this way A + B will have to be equal to zero.

A + B = 0; A = -B

On the other hand, the constant value A will have to be equal to the constant value 3.

A = 3

Thus.

A = 3 and B = -3

Once the numerator values ​​for the simple fractions are already defined, the summation is restated.

Where the generic form of telescopic summation has already been achieved. The telescopic series is developed.

Where when dividing by a very large number the result will get closer and closer to zero, observing the convergence of the series to the value 3.

This type of series could not be solved in any other way, due to the infinite number of iterations that define the problem. However, this method, along with many others, frame the branch of study of numerical series, whose objective is to determine the convergence values ​​or define the divergence of said series.

References

1. Infinitesimal calculus lessons. Manuel Franco, Manuel Franco Nicolás, Francisco Martínez González, Roque Molina Legaz. EDITUM, 1994.
2. Integral Calculus: Sequences and Series of Functions. Antonio Rivera Figueroa. Grupo Editorial Patria, Oct 21. 2014.
3. A Course in Calculus and Real Analysis. Sudhir R. Ghorpade, Balmohan V. Limaye. Springer Science & Business Media, Jun 5. 2006.
4. Infinite series. Tomlinson Fort. The Clarendon Press, 1930.
5. Elements of the Theory of Infinite Processes. Lloyd Leroy Smail. McGraw-Hill Book Company, Incorporated, 1923.