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.
Nonetheless, there are many summation methods that respect these manipulations and that do assign a "sum" to Grandi's series. Two of the simplest methods are Cesàro summation and Abel summation.
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