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Definition of a serie and infinite series
Definition of a serie:
Mathematicians usually define a series as a pair of sequences:
the sequence of terms of the series: a0, a1, a2,
... ; and the sequence of partial sums S0, S1, S2,
... where
.
The notation :
represents then
a
priori this pair of sequences, which is always well defined, but which may or may not converge. In the case of convergence, i.e., if the sequence of partial sums SN has a limit, the notation is also used to denote the limitof this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may sometimes omit the limits (atop and below the sum's symbol) in the former case, although it is usually clear from the context which one is meant. Also, different notions of convergence of such a sequence do exist (absolute convergence, summability, etc). In case the elements of the sequence (and thus of the series) are not simple numbers, but, for example, functions, still more types of convergence can be considered (pointwise convergence, uniform convergence, etc.).
Infinite Series:
The sum of an infinite series a0, a1, a2, ... is the limit of the sequence of partial sums
as n → ∞. This limit can have a finite value; if it does, the series
is said to converge; if it does not, it is said to diverge.
The simplest convergent infinite series is perhaps:
