Main Concept

A series is the sum of terms in a sequence. There are two types of series: finite and infinite. Finite series have defined first and last terms, and infinite series continue indefinitely.

A finite series can be written as follows:

$\sum _{k \=1}^n{x}_{k }\=x_1\+x_2\+x_3\+..\.\+ x_k$, where $x_k\in \mathrm{\ℝ}$ for every $k\=1..n$

An infinite series can be written as follows:

$\sum _{k \=1}^{\infty }{x}_{k }\=x_1\+x_2\+x_3\+..\.$, where $x_k\in \mathrm{\ℝ}$ for every $k\in \mathrm{\ℕ}$

A geometric series is a series in which the term $x_{k\+1}$ can be obtained from the previous term $x_k$ by multiplying by a fixed number. For instance, $\sum _{n\=1}^{\mathrm{\∞}}\frac{1}{2^n}$ is a geometric series in which each successive term is found by multiplying the previous term by $\frac{1}{2}$.

Sometimes the terms of an infinite series can be added up to give a finite number, called the sum of the series.

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