Crazy Math: Infinite Series Has Three Different Results

If you're a math fan and love to solve problems, you'll love the video posted on YouTube by Dr. James Grime from Numberphile. This is a demonstration of the Grandi Series, developed by the Italian philosopher and mathematician Guido Grandi in the 18th century and consisting of an infinite series in which the number "1" is subtracted and added successively.

However, while this story of adding and subtracting number 1 repeatedly and until infinity seems simple - and the answer seems quite obvious - there are three different solutions to this problem, depending on the magic rules you have. adopt to face the challenge. Check out the following video:

Although the video is in English, you can enable relatively understandable Portuguese subtitles in the menu.

As you can see, the series consists of calculating 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ..., infinitely. However, if we add parentheses to the series - (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) ... - the result is zero . On the other hand, by adding parentheses a little differently and adding their contents - 1 + (- 1 + 1) + (- 1 + 1) + (- 1 + 1) + (- 1 ... -, then the answer will be 1 .

However, there is still a third answer, which is the most surprising: let's say the result of the infinite series calculation is a number represented by the letter " S ". So how about we make the 1 - S count, which means the same as 1 - the result of the infinite count? Writing the calculation, it would look like this: 1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ...) .

However, if we delete the parentheses, the minus sign causes all signs to be reversed, so the account becomes: 1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + 1 - 1. ..) = 1 - 1 + 1 - 1 + 1 –..., which is exactly equal to Grandi's infinite series, that is, equal to S again. However, rewriting the calculation we have 1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - 1 + 1 - 1 + 1 –... = S, which is the same as 2S = 1 .

That is, the result of the sum and infinite subtraction of the number 1 according to this third solution is equal to ½ ! To learn more about this calculation and other series with strange results, check out Dr. Grime's explanations in the video.

* Originally posted on 27/06/2013.

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