Cesàro mean

The Cauchy's limit theorem was first formulated by the French mathematician Augustin Louis Cauchy. He is a special case of the general theorem of Cesaro - pride and says: From the convergence of a sequence of numbers, the convergence of the Cesaro means of the sequence to the same limit follows.

Mathematical formulation of the theorem

Given a sequence of numbers, then the sequence of Cesaro means of this sequence is defined by. It follows then.

Related results and extensions

Considering, instead of the ordinary arithmetic average, a weighted average, as follows from the convergence of the original sequence, the convergence of weighted mean, which means that it is the following set:

Be an arbitrary sequence and with a sequence of positive numbers with, then applies.

Also applies an analogous theorem for the geometric mean:

Let be a sequence with, converges also the consequence of the geometric mean, ie

Evidence

Be arbitrary and chosen so that applies to everyone. Because there is one with

For.

For all followed