0 or 1
When the outcome of a calculation is 0 or 1 you should better check all steps once more. With both of these numbers you must always be very careful.
Example 1
The derivative of the inverse tangent can be written as a power series
and also as
so applies
It follows
and if we don't care, we may find
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Let us check this carefully. For x = 0 you get here
and that is of course not correct. Now we take x = 1, and you see
but that is also wrong. If we substitute x = 0 in the original formula we get
and that is correct. We didn't care and replaced 1 – 1 + 1 – 1 + ··· just by 0, allthough this is undefined. That coused the troubles. To be sure, we substitute now x = 1 in the original formula
and that is not correct. So we made an error before. That is the case, as the power series for the inverse tangent is only valid for |x| < 1.