Grandi's series

In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}}$

is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that the sequence of partial sums of the series does not converge.

However, though it is divergent, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Cesàro summation and the Ramanujan summation of this series are both 1/2.