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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
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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.