To prove that one equals zero take an infinite series, for example, 1, -1, +1, -1, +1, ... and group the terms in two different ways, and add them in this way:

Of course this does not prove that one equals zero: it proves that care must be taken with infinite series. Next, take another infinite series in which the terms may be regrouped into any order without fear of paradoxical results. Let a pie be divided into halves. Let one of the halves be divided in two, giving two quarters. Then let one of the two quarters be divided into two eighths. Continue likewise. The infinite series is 1/2, 1/4, 1/8, 1/16, .... No matter how the pieces are rearranged, they should all fit back into the pie plate and exactly fill it.

The danger of infinite series is not that they have an infinite number
of terms but that they may sum to infinity.
Safety is assured if the sum of the absolute values of the terms
is finite.
Such a series is called
*
absolutely convergent.
*

10/31/1997