In mathematics, a series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, while infinite sequences and series continue on indefinitely.
Given an infinite sequence $\left\{{a}_{n}\right\}$, we write the infinite series as $\sum _{n\=1}^{\infty}{a}_{n}equals;{a}_{1}plus;{a}_{2}plus;{a}_{3}plus;..period;plus;{a}_{n1}plus;{a}_{n}plus;{a}_{nplus;1}plus;..period;$
When adding up only the first n terms of a sequence, we refer to the ${\mathit{n}}^{\mathit{th}}$ partial sum: ${s}_{n}equals;\sum _{iequals;1}^{n}{a}_{i}equals;{a}_{1}plus;{a}_{2}plus;..period;plus;{a}_{n}$
So, there are two sequences associated with any series $\sum {a}_{n}$ :
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$\left\{{a}_{n}\right\}$, the sequence of its terms

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$\left\{{s}_{n}\right\}$, the sequence of its partial sums

A series is said to converge if the sequence of its partial sums, $\left\{{s}_{n}\right\}$, converge. The finite limit of ${s}_{n}$ as n approaches infinity is then called the sum of the series:
$Sequals;\underset{n\to \infty}{lim}{s}_{n}equals;\sum _{nequals;1}^{\infty}{a}_{n}$.
This means that by adding sufficiently many terms of the series, we can get very close to the value of S. If $\left\{{s}_{n}\right\}$ diverges, then the series diverges as well.
Finding S is often very difficult, and so the main focus when working with series is often just testing to figure out whether the series converges or diverges.
