REVISION NOTES
IGCSE Edexcel Further Pure Mathematics
1.5 Series
1.5.1 Use of the ∑ notation
![edexcel_igcse_further pure maths_fpm_topic 05_series_013_ geometric sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_013_-geometric-sequence.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_014_sigma notation.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_014_sigma-notation.png)
Find sum when lower limit is NOT 1:
Supper limit – Slower limit – 1
1.5.2 Arithmetic and geometric series
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers that increase or decrease by the same amount. This amount is called common difference.
![edexcel_igcse_further pure maths_fpm_topic 05_series_001_arithmetic sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_001_arithmetic-sequence.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_002_arithmetic sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_002_arithmetic-sequence.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_003_arithmetic sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_003_arithmetic-sequence.png)
Arithmetic Series
An arithmetic series is the sum of the terms in an arithmetic sequence.
![edexcel_igcse_further pure maths_fpm_topic 05_series_004_arithmetic series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_004_arithmetic-series.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_005_arithmetic series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_005_arithmetic-series.png)
Geometric Sequence
A geometric sequence is a sequence of numbers that increase or decrease by a common ratio.
![edexcel_igcse_further pure maths_fpm_topic 05_series_006_geometric sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_006_geometric-sequence.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_007_geometric sequence.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_007_geometric-sequence.png)
Geometric Series
An geometric series is the sum of the terms in an geometric sequence.
Type 1: Geometric Series (Finite)
![edexcel_igcse_further pure maths_fpm_topic 05_series_008_finite geometric series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_008_finite-geometric-series.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_009_finite geometric series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_009_finite-geometric-series.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_010_finite geometric series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_010_finite-geometric-series-1.png)
Type 2: Geometric Series (Infinite)
If |r| < 1, then rn approaches 0 for very large n.
We say that the series converges. We call this the limiting sum of the series.
![edexcel_igcse_further pure maths_fpm_topic 05_series_011_infinite geometric series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_011_infinite-geometric-series.png)
![edexcel_igcse_further pure maths_fpm_topic 05_series_012_infinite geometric series.png](https://resource.studiaacademy.com/wp-content/uploads/2022/05/edexcel_igcse_further-pure-maths_fpm_topic-05_series_012_infinite-geometric-series.png)