Edexcel IAL - Further Pure Mathematics 2- 2.1 Summation of Finite Series by Differences- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 2- 2.1 Summation of Finite Series by Differences -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 2- 2.1 Summation of Finite Series by Differences -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 2.1 Summation of Finite Series by Differences
Summation of Finite Series Using the Method of Differences
The method of differences (also called a telescoping method) is used to sum series in which many terms cancel out when written in a suitable form.
This method is especially useful for series involving rational expressions.
Key Idea
A term such as
\( \dfrac{1}{r(r+1)} \)
can be written using partial fractions as:
\( \dfrac{1}{r(r+1)} = \dfrac{1}{r} – \dfrac{1}{r+1} \)
When these terms are added in a series, most terms cancel.
Example of Telescoping
Consider:
\( \sum_{r=1}^{n} \left(\dfrac{1}{r} – \dfrac{1}{r+1}\right) \)
Write out the first few terms:
\( \left(1 – \dfrac{1}{2}\right) + \left(\dfrac{1}{2} – \dfrac{1}{3}\right) + \left(\dfrac{1}{3} – \dfrac{1}{4}\right) + \dots + \left(\dfrac{1}{n} – \dfrac{1}{n+1}\right) \)
All the middle terms cancel, leaving:
\( 1 – \dfrac{1}{n+1} \)
General Result
Therefore,
\( \sum_{r=1}^{n} \dfrac{1}{r(r+1)} = 1 – \dfrac{1}{n+1} = \dfrac{n}{n+1} \)
Example
Find \( \sum_{r=1}^{5} \dfrac{1}{r(r+1)} \).
▶️ Answer / Explanation
\( \sum_{r=1}^{5} \dfrac{1}{r(r+1)} = \dfrac{5}{6} \)
Example
Find \( \sum_{r=1}^{n} \dfrac{2}{r(r+1)} \).
▶️ Answer / Explanation
\( \sum_{r=1}^{n} \dfrac{2}{r(r+1)} = 2 \sum_{r=1}^{n} \left(\dfrac{1}{r} – \dfrac{1}{r+1}\right) \)
\( = 2\left(1 – \dfrac{1}{n+1}\right) = \dfrac{2n}{n+1} \)
Example
Find \( \sum_{r=1}^{n} \dfrac{3}{(r+1)(r+2)} \).
▶️ Answer / Explanation
First use partial fractions:
\( \dfrac{3}{(r+1)(r+2)} = \dfrac{3}{r+1} – \dfrac{3}{r+2} \)
So
\( \sum_{r=1}^{n} \left(\dfrac{3}{r+1} – \dfrac{3}{r+2}\right) \)
This telescopes to:
\( \dfrac{3}{2} – \dfrac{3}{n+2} \)