The Arithmetic field of study covers any activity that studies mathematics in order to manifest the material. The Calculus field of study covers any activity that studies Calculus in order to manifest the material. In mathematics, a series is often represented as the sum of a sequence of terms. That is, a series is represented as a list of numbers with addition operations between them, for example this arithmetic sequence:

1 + 2 + 3 + 4 + 5 + ... + 99 + 100

In most cases of interest the terms of the sequence are produced according to a certain rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator. A series may be finite or infinite. Finite series 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. Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:

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

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

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

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