Grandi series

It is called the Grandi series. Sherry; He was dedicated to Italian mathematician, philosopher and pastor Guido Grandi for his original works in 1703. In general terms, the sum of the expression defined as a divergent series without sum is ½.
The simplest way to calculate the sum of 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + bir is to detect it as a nested series and to perform subtraction operations directly. (1 - 1) + (1 - 1) + (1 - 1) + yolla = 0 + 0 + 0 +… = 0 On the other hand, when the terms are clustered in a different way, the total contradicts the result obtained above. 1 + (değer1 + 1) + (yoluyla1 + 1) + (raç1 + 1) + 1 = 1 + 0 + 0 + 0 + 1 = 1 "values" 0 and 1, which can be accessed by separating the Grandi series with the help of separators. 'dr. Eilenberg zaman A similar method, called the Mazur trick, is used from time to time in knot theory and algebra. Considering the Grandi series as a divergent geometric series, the methods applied to the convergent geometric series can be adapted to this series and a different value can be found. S = 1 - 1 + 1 - 1 + ⁄ and 1 - S = 1 - (1 - 1 + 1 - 1 + +) = 1 - 1 + 1 - 1 + S = SS = 1…2 The same result is calculated as --S the result can also be achieved with the 2S = 1 solution after subtracting it from S. These changes on the series do not focus on exactly what the sum of a series means. Although it is important to divide the series into groups and perform four operations on them, it is possible to reach the following two results: There is no sum of 1 - 1 + 1 - 1 + ayır series. ... but total 1 ...2. Both statements can be verifiable and verifiable, but to accomplish this, mathematical concepts found in the 19th century are needed. The period from the arrival of calculus to Europe until the end of the 18th century witnessed the "never-ending" and "hard" discussions among mathematicians.

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