Introduction to Series

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Introduction to Series

Sequences

If you have a set of numbers T1, T2, T3,...where there is a rule for working out the next numbers, we call this set a sequence.

Every number in the sequence is called a term and Tn is the nth term.

Examples:

b) 3, 9, 27, 81... nth term = 3n

a) 3, 6, 9, 12...nth term = 3n

c) 1, 3, 6, 10... nth term = ½n(n+1) (the triangle numbers)

Series

A series is just when the terms of a sequence are added: T1 + T2 + T3 ... Tn

For example: 1 + 2 + 4 + 8 + ...

A finite series stops after a certain number of terms,

For example: 1 + 3 + 9 + 27 + 81, which has five terms.

An infinite series does not stop,

Infinite series

Sigma notation

We use the symbol ∑ to define 'the sum of the terms' so that:

Example

is the sum of all the terms where t takes the values between 2 and 6 inclusive.

Written out in full, this is:

Example

Another example:

Example

The first term is when t = 3 i.e. 2 × 3 − 6 = 0.

The last tern is when t = 12 i.e. 2 × 12 − 6 = 18.

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