Summation of Grandi's series

The formal manipulations that lead to 1 − 1 + 1 − 1 + · · · being assigned a value of 1⁄2 include: * Adding or subtracting two series term-by-term, * Multiplying through by a scalar term-by-term, * "Shifting" the series with no change in the sum, and * Increasing the sum by adding a new term to the series' head. These are all legal manipulations for sums of convergent series, but 1 − 1 + 1 − 1 + · · · is not a convergent series.

Summation of Grandi's series

The formal manipulations that lead to 1 − 1 + 1 − 1 + · · · being assigned a value of 1⁄2 include: * Adding or subtracting two series term-by-term, * Multiplying through by a scalar term-by-term, * "Shifting" the series with no change in the sum, and * Increasing the sum by adding a new term to the series' head. These are all legal manipulations for sums of convergent series, but 1 − 1 + 1 − 1 + · · · is not a convergent series.