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Grandi's series

There is no solution for the infinite Grandi's series

1 − 1 + 1 − 1 + 1 − 1 + ··· = ?

How can this occur?

 


Example 1

The first derivative of the inverse tangent is

The series for the inverse tangent is

and the derivative of it gives

So we write the equation

It follows

and if we don't mind the details we will find

          

We better check that carefully. Let x = 0 and you get

          

what is of course incorrect. Now we take x = 1, so we see

          

and that is also wrong. If we substitute x = 0 in the orginal equation, we find

which is correct. We forgot the details, and instead of 1 + 1 – 1 + 1 – 1 ···  we wrote just 0, although this is undefined. That caused the troubles. As a final check we substitute x = 1 in the orginal equation

         

and that is not correct. We apparently have already made a mistake at the very beginning. And that is the case, as the series vor the inverse tangent is only valid for x | < 1.

 


Additional information

The Italian mathematician Guido Grandi wrote about this series in1703.

 


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