Series (mathematics)

In mathematics, a series is the sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them, e.g,

Related Topics:
Mathematics - Sum - Sequence - Term

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:1 + 2 + 3 + 4 + 5 + ...

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

which may or may not be meaningful.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.

Related Topics:
Formula - Algorithm - Measurement - Random number generator

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.

Related Topics:
Finite - Algebra - Mathematical analysis

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

Related Topics:
Arithmetic series - Arithmetic progression

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sum_{n=0}^k (an+b);

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

and finite geometric series, a sum of a geometric progression, which can be written as:

Related Topics:
Geometric series - Geometric progression

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sum_{n=0}^k a^{n}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~