Grandi's series ($1-1+1-1+\cdots$)

The series with terms $a_n=(-1{)}^n$, indexed from zero, named for Guido Grandi. Its even-indexed partial sums are one and its odd-indexed partial sums are zero, so it has no ordinary sum. First-order Cesàro summation gives $1/2$. Grouping adjacent pairs would obscure this distinction: the partial sums of the grouped expressions select only one of the original subsequences.