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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.