Series – iGCSE Further Pure Mathematics Edexcel Revision

Introduction to Series

A series is the sum of the terms of a sequence.
Finite series have a specific number of terms, whereas infinite series continue indefinitely.
The nth term of a sequence can be found using various methods, such as studying the difference between consecutive terms, or by using the method of differences for more complex sequences.

An arithmetic series is one where the difference between each consecutive pair of terms is constant. This constant is called the common difference.
The nth term of an arithmetic series can be calculated with the formula: a + (n - 1)d, where a is the first term, d is the common difference and n is the term number.
The sum, Sn, of the first n terms of an arithmetic series can be found using the formula: Sn = n/2 (2a + (n - 1)d).

A geometric series is one where each consecutive pair of terms have a constant ratio. This constant is called the common ratio.
The nth term of a geometric series can be calculated with the formula: ar^(n - 1), where a is the first term, r is the common ratio and n is the term number.
The sum, Sn, of the first n terms of a geometric series can be found using the formula: Sn = a(1 - r^n) / (1 - r) if r < 1.

Summation notation, also known the sigma notation, is a way of writing long summations in a condensed form.
The Greek letter, Σ (sigma), is used to denote the sum of a series.
For instance, the sum of the first n terms a series can be written Σ(from i=1 to n) of a_i, where a_i refers to each term of the series.

An infinite series is said to converge if the sum of the series approaches a certain value as the number of terms tends to infinity. Likewise, a series is said to diverge if the sum does not approach a certain value.
A geometric series converges if the absolute value of the common ratio, r , is less than 1 and diverges otherwise.
The sum to infinity, S∞, of a converging geometric series can be calculated using the formula: S∞ = a / (1 - r).