What is the difference between summing a series using standard formulae and using the method of differences?

Standard formulae are used when the general term is a polynomial in $r$ (e.g., $r^3 + 2r$). The method of differences is used for terms that can be written as a difference, like $f(r) - f(r+1)$, which is common for rational functions (fractions with polynomials).

How do I handle a sum that doesn't start at r=1, for example, $\sum_{r=5}^{10} u_r$?

You use the principle of inclusion-exclusion. Calculate the sum from 1 to the upper limit and subtract the sum from 1 to the term before the lower limit. So, $\sum_{r=5}^{10} u_r = \sum_{r=1}^{10} u_r - \sum_{r=1}^{4} u_r$. This applies to both standard formulae and the method of differences.

What if I'm given a term for the method of differences but it's not in the form $f(r) - f(r+1)$?

The question will usually guide you. Often, you'll first be asked to 'Show that' the term can be expressed using partial fractions, which naturally creates the difference form. If you have to find it yourself, look for patterns. For example, $r(r+1)!$ can be written as $(r+1-1)(r+1)! = (r+2)! - (r+1)!$.

Can I just put the formulae for $\sum r, \sum r^2, \sum r^3$ into my calculator to find the sum?

While a calculator can compute a numerical sum for a given $n$, exam questions almost always require you to find an algebraic expression for $S_n$ in terms of $n$. You must show the substitution of the standard formulae and the subsequent algebraic simplification and factorisation to earn the marks.

Where can I learn Summation of series for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 1.3 (Summation of series) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/1-3-summation-of-series.

Is Summation of series (1.3) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Summation of series.

Summation of series 9231 — Free A-Level Further

Summation of series

Welcome to the study of summing series. While you're familiar with arithmetic and geometric progressions, many other series exist. In this topic, we'll learn powerful, systematic methods to find the sum of series whose general term is a polynomial, and a clever technique called the 'method of differences' for other types of series.

BY THE END, YOU CAN…

1Recall and use the standard results for the sums of the first n integers, squares, and cubes.
2Apply the method of differences to find the sum of a finite series.
3Determine if a series converges and, if so, find its sum to infinity.
4Solve problems involving the summation of series in various contexts.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Domino Demolition

The method of differences is a clever way to sum a complicated series by making most of its terms cancel out. This leaves only a few terms at the beginning and end, making the final calculation simple.

THINK OF IT LIKE…

Imagine setting up a very long line of dominoes. When you topple the first one, it knocks over the next, which knocks over the next, and so on. If you want to know the overall result, you don't need to watch every single domino fall. You just need to see the first one get pushed and the last one hit the table. The intermediate dominoes are just part of the chain reaction that cancels itself out. The method of differences works the same way: the middle terms of the sum cancel each other out, leaving only the 'first push' and the 'final fall'.

1
Express the general term of the series, $u_r$ , in the form $f(r) - f(r+1)$ . This is often done using partial fractions.
2
Write out the first few terms of the sum (e.g., for r=1, r=2, r=3) and the last few terms (e.g., for r=n-1, r=n).
3
Observe the 'telescoping' cancellation. The $+f(r+1)$ part of one term cancels with the $-f(r)$ part of the next.
4
Add together the remaining terms that did not cancel. This will give you the formula for the sum to n terms, $S_n$ .

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1) \\ \\ \sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1) \\ \\ \sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$

Tap a symbol — great for exam definitions

$u_r = f(r) - f(r+1)$

Tap a symbol — great for exam definitions

$S_{\infty} = \lim_{n \to \infty} S_n$

Full topic notes

Formal explanation with the rigour you need for the exam.

Standard Summation Formulae

For series where the general term is a polynomial in $r$ , we can find the sum by using a set of standard results. These formulae, provided in the formula book, allow us to sum the first $n$ integers, their squares, and their cubes. By combining these, we can sum any polynomial.

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1) \\ \\ \sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1) \\ \\ \sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$

To sum a polynomial like $\sum (ar^3 + br^2 + cr + d)$ , split it up: $a\sum r^3 + b\sum r^2 + c\sum r + \sum d$ .
Remember that $\sum_{r=1}^{n} d = nd$ .
If a sum does not start at $r=1$ , for example $\sum_{r=a}^{n} u_r$ , calculate it as $\sum_{r=1}^{n} u_r - \sum_{r=1}^{a-1} u_r$ .

The Method of Differences

This powerful technique is used when the general term of a series, $u_r$ , can be expressed as the difference between two consecutive terms of another sequence, $f(r)$ . When we write out the terms of the sum, most of them cancel out in a chain reaction, leaving just a few terms at the beginning and end. This is also known as a telescoping series.

If $u_r = f(r) - f(r+1)$ , then $\sum_{r=1}^{n} u_r = [f(1)-f(2)] + [f(2)-f(3)] + \dots + [f(n)-f(n+1)] = f(1) - f(n+1)$

The first step is usually to express $u_r$ in the required form. For rational functions, this often involves partial fractions.
Always write out the first two or three terms and the last two terms to be certain which terms cancel and which remain.
The form might not always be $f(r) - f(r+1)$ . It could be $f(r+1) - f(r)$ , or even $f(r) - f(r+2)$ , which would leave more terms at the start and end.

Sum to Infinity

For some series, as we add more and more terms, the sum approaches a specific finite value. This value is called the 'sum to infinity', denoted $S_{\infty}$ . We can find this by evaluating the limit of the expression for the sum of the first $n$ terms, $S_n$ , as $n$ tends to infinity. If the limit is a finite number, the series is said to converge. If the sum grows without bound, the series diverges.

$S_{\infty} = \lim_{n \to \infty} S_n$

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Find the sum of the series $\sum_{r=1}^{n} (r^2 - 3r + 2)$ . Express your answer in fully factorised form.

1
First, we split the summation into parts based on the standard formulae:

EXAMPLE 2

i) Express $\frac{2}{(2r-1)(2r+1)}$ in partial fractions.

ii) Hence, find an expression for $S_n = \sum_{r=1}^{n} \frac{2}{(2r-1)(2r+1)}$ .

iii) Find the sum to infinity of the series.

1
i) Let $\frac{2}{(2r-1)(2r+1)} \equiv \frac{A}{2r-1} + \frac{B}{2r+1}$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEASummation of series

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is the formula for $\sum_{r=1}^{n} r$ ?

TAP OR SPACE TO FLIP

ANSWER

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)$

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
To sum a polynomial like $\sum (ar^3 + br^2 + cr + d)$ , split it up: $a\sum r^3 + b\sum r^2 + c\sum r + \sum d$ .
✓
Remember that $\sum_{r=1}^{n} d = nd$ .
✓
If a sum does not start at $r=1$ , for example $\sum_{r=a}^{n} u_r$ , calculate it as $\sum_{r=1}^{n} u_r - \sum_{r=1}^{a-1} u_r$ .

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.2 Rational functions and graphs NEXT →1.4 Matrices

Summation of series

BY THE END, YOU CAN…

1Recall and use the standard results for the sums of the first n integers, squares, and cubes.
2Apply the method of differences to find the sum of a finite series.
3Determine if a series converges and, if so, find its sum to infinity.
4Solve problems involving the summation of series in various contexts.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Domino Demolition

THINK OF IT LIKE…

1
Express the general term of the series, $u_r$ , in the form $f(r) - f(r+1)$ . This is often done using partial fractions.
2
Write out the first few terms of the sum (e.g., for r=1, r=2, r=3) and the last few terms (e.g., for r=n-1, r=n).
3
Observe the 'telescoping' cancellation. The $+f(r+1)$ part of one term cancels with the $-f(r)$ part of the next.
4
Add together the remaining terms that did not cancel. This will give you the formula for the sum to n terms, $S_n$ .

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1) \\ \\ \sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1) \\ \\ \sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$

Tap a symbol — great for exam definitions

$u_r = f(r) - f(r+1)$

Tap a symbol — great for exam definitions

$S_{\infty} = \lim_{n \to \infty} S_n$

Full topic notes

Formal explanation with the rigour you need for the exam.

Standard Summation Formulae

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1) \\ \\ \sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1) \\ \\ \sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n+1)^2$

To sum a polynomial like $\sum (ar^3 + br^2 + cr + d)$ , split it up: $a\sum r^3 + b\sum r^2 + c\sum r + \sum d$ .
Remember that $\sum_{r=1}^{n} d = nd$ .
If a sum does not start at $r=1$ , for example $\sum_{r=a}^{n} u_r$ , calculate it as $\sum_{r=1}^{n} u_r - \sum_{r=1}^{a-1} u_r$ .

The Method of Differences

If $u_r = f(r) - f(r+1)$ , then $\sum_{r=1}^{n} u_r = [f(1)-f(2)] + [f(2)-f(3)] + \dots + [f(n)-f(n+1)] = f(1) - f(n+1)$

The first step is usually to express $u_r$ in the required form. For rational functions, this often involves partial fractions.
Always write out the first two or three terms and the last two terms to be certain which terms cancel and which remain.
The form might not always be $f(r) - f(r+1)$ . It could be $f(r+1) - f(r)$ , or even $f(r) - f(r+2)$ , which would leave more terms at the start and end.

Sum to Infinity

$S_{\infty} = \lim_{n \to \infty} S_n$

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Find the sum of the series $\sum_{r=1}^{n} (r^2 - 3r + 2)$ . Express your answer in fully factorised form.

1
First, we split the summation into parts based on the standard formulae:

EXAMPLE 2

i) Express $\frac{2}{(2r-1)(2r+1)}$ in partial fractions.

ii) Hence, find an expression for $S_n = \sum_{r=1}^{n} \frac{2}{(2r-1)(2r+1)}$ .

iii) Find the sum to infinity of the series.

1
i) Let $\frac{2}{(2r-1)(2r+1)} \equiv \frac{A}{2r-1} + \frac{B}{2r+1}$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEASummation of series

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is the formula for $\sum_{r=1}^{n} r$ ?

TAP OR SPACE TO FLIP

ANSWER

$\sum_{r=1}^{n} r = \frac{1}{2}n(n+1)$

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
To sum a polynomial like $\sum (ar^3 + br^2 + cr + d)$ , split it up: $a\sum r^3 + b\sum r^2 + c\sum r + \sum d$ .
✓
Remember that $\sum_{r=1}^{n} d = nd$ .
✓
If a sum does not start at $r=1$ , for example $\sum_{r=a}^{n} u_r$ , calculate it as $\sum_{r=1}^{n} u_r - \sum_{r=1}^{a-1} u_r$ .

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.2 Rational functions and graphs NEXT →1.4 Matrices

Revise Summation of series for Cambridge 9231 (1.3)

In simple terms

Explore the concept

Key formulas

Full topic notes

Standard Summation Formulae

The Method of Differences

Sum to Infinity

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Standard Summation Formulae

The Method of Differences

Sum to Infinity

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Summation of series