Averaging the partial sums is the simplest case of what's known as Cesàro sums. Here's why that technique is attractive: If the original series converges to a particular value, so does its Cesàro sum; if the original series diverges, so does its Cesàro sum. For series that are neither convergent nor divergent, such as the Grandi series, sometimes the Cesàro sum produces a definite finite value. Some people (not I!) therefore think that the Cesàro sum can be used in place of the standard sum.

Averaging the partial sums is the simplest case of what's known as Cesàro sums. Here's why that technique is attractive: If the original series converges to a particular value, so does its Cesàro sum; if the original series diverges, so does its Cesàro sum. For series that are neither convergent nor divergent, such as the Grandi series, sometimes the Cesàro sum produces a definite finite value. Some people (not I!) therefore think that the Cesàro sum can be used in place of the standard sum.