An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form
$${\displaystyle a_{0}+a_{1}+a_{2}+\cdots ,}$$ See more

Partial summation takes as input a sequence, (an), and gives as output another sequence, (SN). It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted … See more

There exist many tests that can be used to determine whether particular series converge or diverge.
• n-th term test: If $${\textstyle \lim _{n\to \infty }a_{n}\neq 0}$$, then the series diverges; if $${\textstyle \lim _{n\to \infty }a_{n}=0}$$ See more

• A geometric series is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: See more

Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, … See more

A series of real- or complex-valued functions
$${\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}$$
converges pointwise on … See more

Development of infinite series
Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of … See more