Grandi's series

Remarks

Several summation methods assign it the value $1/2$. Cesàro summation: the averages $(s_0 + \cdots + s_N)/(N+1)$ equal $1/2$ for odd $N$ and $(N/2 + 1)/(N+1)$ for even $N$, so they tend to $1/2$. Abel summation: for $0 < x < 1$, $\sum_{n \ge 0} (-1)^n x^n = 1/(1 + x)$, which tends to $1/2$ as $x \to 1^-$. Borel summation also gives $1/2$, since it sums the geometric series $\sum z^n$ to $1/(1 - z)$ at $z = -1$. The value $1/2$ is forced for any linear and stable method that sums the series: if $S = 1 - 1 + 1 - \cdots$, stability and linearity give $S = 1 - S$.