### Series and Sequences

#### Definitions

##### Sequence

A sequence is a list of numbers which are written in a particular order. For example, $1, 3, 5, 7, 9$ is a sequence of odd numbers. The algebraic notation used for sequences is $u_1$ for the first term, $u_2$ for the second term, and so on. The $n^{\text{th} }$ term in a sequence is denoted $u_n$.

A finite sequence or simple sequence is a finite list of numbers, for example $1, 3, 5, \dotso, 19$.

An infinite sequence is like a simple sequence, but the terms go on forever, for example $2, 5, 8, \dotso$.

The 'three dots' notation stands in for missing terms. If the dots are followed by a final number, the sequence is finite. If the dots have nothing after them, the sequence is infinite.

##### Series

A series is obtained from a sequence by adding the terms together. A series is denoted by $S_n = u_1+u_2+u_3+\dotso +u_n$

The Greek capital letter sigma, $\sum$, is used to denote the sum of a series. The starting index is written underneath and the final index above, and the series to be summed is written on the right. A series does not have to be the sum of all the terms in a sequence. For example,

$\sum^{5}_{k=1}{2^k}$ represents the sum of the first five terms in the sequence of powers of $2$.

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