Series (mathematics)

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely. To emphasize that there are an infinite number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, the value of a series is assigned by evaluation of a limit of a sequence of terms.

Given an infinite sequence ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$, the associated series is the expression obtained by adding all those terms together: ${\displaystyle a_{1}+a_{2}+a_{3}+\cdots }$. These can be written more compactly as ${\displaystyle \textstyle \sum _{i=1}^{\infty }a_{i},}$ by using the summation symbol ${\textstyle \sum }$, a capital Greek letter sigma. The sequence can be composed of any kind of mathematical object for which addition is defined and there is a notion of convergence.

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