Summable Infinite Series

The basic idea here is to try to use another limiting process to form the sum of an infinite series when the limit of the sequence of partial sums fails to exist.

This second approach, the method of "summation by Cesaro means", will always yield the ordinary sum (as the limit of the sequence of partial sums) for an infinite series that converges.

The definition

Let

u₁ + u₂ + u₃ + ...

be a given infinite series of real or complex numbers.

Let

s_n = u₁ + u₂ + ... + u_n

be the n^th partial sum of the given series.

Let

t_n = (1/n)(s₁ + s₂ + ... + s_n)

Definition: The infinite series is summable by Cesaro means if

t = lim_{{n --> infinity}} t_n

exists. If this limit exists, it is called the sum of the series.

Example

We have to look no further than the geometric series to find an excellent illustration.

Let z be a complex number. The geometric series is the series

1 + z + z² + z³ + z⁴ + ...

whose terms are the non-negative powers of z.

We have seen (cf. the text) that this series converges to the sum

1/(1-z)

when |z| < 1 and diverges for all other values of z.

This series is summable when |z| = 1, other than for z = 1, with sum

1/(1-z) .

Note that one could not hope to have 1/(1-z) be the sum when z = 1 since that would involve a division by zero.

Proof.

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