Finite Associativity and Infinite Sums: The Case of Grandi's Series

The finite identity (a+b)+c=a+(b+c) holds without exception in the ring of integers. However, it does not automatically justify removing or modifying parentheses in an infinite sum. The reason is not a failure of algebraic associativity, but the fact that an infinite series is not defined by infinitely many applications of the binary operation of addition. Instead, it is defined as the limit of the sequence of partial sums. This paper rigorously analyses the expression 0+0+0+...=(1-1)+(1-1)+(1-1)+... and explains why the series of blocks converges to zero, whereas the flattened series 1-1+1-1+... diverges in the ordinary sense. A general criterion is also presented to identify the information lost when a series is grouped into finite consecutive blocks. The distinction between ordinary convergence and Cesàro summability is discussed through Grandi’s series.