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Sequences and Series

A sequence is a collection of numbers in a defined order. The terms of a sequence follow a certain rule.

E.g. 2, 5, 8, 11, .... rule: start at 2 and add 3 to each consecutive number. This is an arithmetic sequence.

0, 3, 6, 12, 24, .... rule: start at 3 and double each term to produce the next term. This is a geometric sequence.

The sum of all the terms in a sequence is a series. A series can have a finite or infinite number of terms.

$1 + 1/2 + 1/3 + 1/4 + 1/5$ .... is the harmonic series.

The sum of a series is symbolised by the capital Greek letter sigma (Σ). Limits may be set for a finite sum:

$$∑↙{i=1}↖n 1/i$$

This is shorthand for: $1/1 + 1/2 + 1/3 + ... + 1/n$

Leonard Euler was the first to use this notation for series.

Arithmetic sequences and series

A sequence or series which has a constant common difference, d, between two consecutive terms is arithmetic, with the general term:

$$u_n = u_{1} + (n - 1)d$$

where $u_1$ is the first term and d is the common difference.

The sum of a finite arithmetic series is:

$$S_n = n/2[2u_1 + (n - 1)d] = n/2(u_1 + u_n)$$

where n is the number of terms in the series, $u_1$ amd $u_d$ are the first and last terms, and each term is separated by a common difference, d.

Geometric sequences and series

If the ratio of two consecutive terms in a sequence or series is constant, then it is geometric.

For the sequence 1, 3, 9, 27, ... ; $u_1 = 1$ is the first term, and r = 3 is the common ratio. The recursive equation is: $u_n = u_{n-1}⋅r$.

The general term is: $u_n = u_1⋅r^{n-1}$, r ≠ -1, 0, 1.

The sum of a geometric series is:

$$S_n = {u_1(1 - r^n)}/{1 - r}, r ≠ 1$$

where n is the number of terms in the series, $u_1$ the first term, and r is the common ratio of any two consecutive terms.

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