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# Introduction to Series

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## Introduction to Series

**Sequences **

If you have a set of numbers T_{1}, T_{2}, T_{3},...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 T_{n} is the **nth** term.

**Examples:**

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

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: T_{1} + T_{2} + T_{3} ... T_{n}

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,

**Sigma notation**

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

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

Written out in full, this is:

Another 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.