Cesàro summation

Cesàro summation assigns a generalized value to a series by taking the limit of the arithmetic means of its partial sums. It agrees with ordinary summation whenever the series converges.

Partial sum: The sum of the first finitely many terms of a sequence or series. Cesàro summation averages the successive partial sums rather than the original terms.
Ernesto Cesàro: An Italian mathematician who introduced the summability method now bearing his name. His work gave the averaging procedure its name and early mathematical formulation.
Infinite series: An expression formed by adding the terms of an infinite sequence. Cesàro summation extends the ways an infinite series can be assigned a value.
Grandi's series: The divergent series 1 − 1 + 1 − 1 + ⋯. Its partial sums alternate between 1 and 0, while their averages approach one half.
Abel summation: A method that assigns a series value from the limit of its power series as the variable approaches one. Abel summation and Cesàro summation often agree but use different limiting procedures.
Arithmetic mean: The sum of a finite collection of numbers divided by the number of values. Each Cesàro average is the arithmetic mean of an initial segment of partial sums.
Fourier series: A representation of a periodic function as a sum of sine and cosine terms. Cesàro summation became useful for assigning limits to Fourier series that fail to converge pointwise.
Sequence: An ordered list of terms indexed by integers or another set of indices. The partial sums and their averages are both sequences whose limits can be tested.
Alternating series: A series whose consecutive terms alternate in sign. Alternation can prevent ordinary convergence even when Cesàro averages settle to a limit.
Borel summation: A method that transforms a series using factorial scaling and an exponential integral. It can sum some divergent series by a transformation unlike Cesàro averaging.

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