When do I use $S_n = \frac{a(1-r^n)}{1-r}$ versus $S_n = \frac{a(r^n-1)}{r-1}$?

Both formulae are correct and will give the same answer. However, to make calculations easier and avoid negative signs, it's conventional to use $S_n = \frac{a(1-r^n)}{1-r}$ when $|r| 1$.

Why is the binomial expansion only valid for $|x|<1$ when n is not a positive integer?

When $n$ is not a positive integer, the expansion is an infinite series. For the sum of this series to converge to a finite value (equal to $(1+x)^n$), the terms must get smaller and smaller. This only happens if $|x|<1$. If $|x| \ge 1$, the terms get larger or stay the same size, and the sum diverges (goes to infinity).

How can I tell if a problem involves an AP or a GP?

Look for keywords. 'Common difference' implies an AP. 'Common ratio' implies a GP. If you're given three consecutive terms, say $x, y, z$: for an AP, $y-x = z-y$ (or $2y = x+z$); for a GP, $y/x = z/y$ (or $y^2 = xz$).

Where can I learn Series for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 1.6 (Series) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/1-6-series.

Is Series (1.6) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Series.

Series 9709 — Free A-Level Maths

Series

Welcome to the study of series. This topic builds on your knowledge of sequences by exploring how to sum their terms. We will cover the two most important types of progressions, arithmetic and geometric, and learn the formulae to efficiently manage them. We will also extend our knowledge of the binomial theorem to cover non-integer powers, a powerful tool for approximation.

BY THE END, YOU CAN…

1Understand and use the binomial expansion of (1+x)ⁿ for any rational n and |x|<1.
2Recognise and work with arithmetic progressions, using formulae for the n-th term and the sum of n terms.
3Recognise and work with geometric progressions, using formulae for the n-th term and the sum of n terms.
4Understand convergence for a geometric series and use the formula for the sum to infinity.

In simple terms

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

Building with Patterns

A series is simply the sum of the terms in a sequence. We'll explore two main types: arithmetic (adding a constant amount) and geometric (multiplying by a constant amount), and learn powerful shortcuts to calculate their sums.

THINK OF IT LIKE…

Imagine two savings plans. Plan A (Arithmetic) is adding a fixed £10 to your piggy bank every week. Plan B (Geometric) is a bank account where your money grows by 1% each month. The first plan grows steadily, while the second grows exponentially. Series formulae help us calculate the total saved after any number of weeks or months without adding it all up manually.

1
Binomial expansion: (1 + x)ⁿ for |x| < 1 when n is not a positive integer.
2
Arithmetic series: u_n = a + (n−1)d; sum S_n = n/2(2a + (n−1)d).
3
Geometric series: u_n = ar^(n−1); sum S_n = a(1−r^n)/(1−r) for r ≠ 1.
4
Sum to infinity exists when |r| < 1: S_∞ = a/(1−r).

Explore the concept

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

INTERACTIVESTEP 1 / 3 · Terms

Each term is the previous one times r: a, ar, ar², …

Key formulas

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

$u_n = a + (n-1)d$

Tap a symbol — great for exam definitions

$u_n = ar^{n-1}$

Tap a symbol — great for exam definitions

$S_\infty = \frac{a}{1-r}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Binomial Expansion for any Rational Index

Previously, you learned to expand $(a+b)^n$ for a positive integer $n$ . Now we extend this to cases where $n$ can be any rational number (e.g., negative integers or fractions). This new expansion is an infinite series and is only valid for a specific range of values.

For any rational $n$ , and for $|x|<1$ :

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

This formula only works when the first term in the bracket is 1.
To expand $(a+bx)^n$ , you must first write it as $a^n(1+\frac{b}{a}x)^n$ .
The expansion is an infinite series, so questions will ask for the first few terms.
The validity condition is crucial: for $(1+kx)^n$ , the expansion is valid when $|kx|<1$ , or $|x| < \frac{1}{|k|}$ .

Arithmetic Progressions (AP)

An arithmetic progression is a sequence where each term after the first is found by adding a constant, called the common difference ( $d$ ), to the previous term. For example, 5, 8, 11, 14, ... is an AP with first term $a=5$ and common difference $d=3$ .

n-th term: $u_n = a + (n-1)d$

Sum of first n terms: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a+l)$

Geometric Progressions (GP)

A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a constant, called the common ratio ( $r$ ). For example, 3, 6, 12, 24, ... is a GP with first term $a=3$ and common ratio $r=2$ .

n-th term: $u_n = ar^{n-1}$

Sum of first n terms: $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$

The sum formula can also be written as $S_n = \frac{a(r^n-1)}{r-1}$ . This form is often more convenient to use when $|r|>1$ as it avoids working with negative numbers in the numerator and denominator, but both forms are mathematically equivalent.

Sum to Infinity of a Geometric Progression

If the common ratio $r$ has a magnitude less than 1 (i.e., $-1 < r < 1$ ), the terms of the GP get progressively smaller. As you add more and more terms, the sum approaches a specific finite value. This is called the 'sum to infinity'. If $|r| \ge 1$ , the sum does not converge and the sum to infinity does not exist.

Sum to infinity: $S_\infty = \frac{a}{1-r}$ , valid only for $|r|<1$ .

Worked examples

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

EXAMPLE 1

i) Find the first three terms in the expansion of $\sqrt{1-4x}$ in ascending powers of $x$ .

ii) State the range of values of $x$ for which the expansion is valid.

1
i) First, rewrite the expression in index form: $\sqrt{1-4x} = (1-4x)^{1/2}$ . This is in the form $(1+y)^n$ with $y = -4x$ and $n = \frac{1}{2}$ .

EXAMPLE 2

The second term of a geometric progression is 12 and its sum to infinity is 50. The common ratio is positive. Find the first term of the progression.

1
Let the first term be $a$ and the common ratio be $r$ .

How it all connects

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

MAIN IDEASeries

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 difference between a sequence and a series?

TAP OR SPACE TO FLIP

ANSWER

A sequence is a list of numbers in a specific order (e.g., 2, 4, 6, 8). A series is the sum of the terms of a sequence (e.g., 2 + 4 + 6 + 8).

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
This formula only works when the first term in the bracket is 1.
✓
To expand $(a+bx)^n$ , you must first write it as $a^n(1+\frac{b}{a}x)^n$ .
✓
The expansion is an infinite series, so questions will ask for the first few terms.
✓
The validity condition is crucial: for $(1+kx)^n$ , the expansion is valid when $|kx|<1$ , or $|x| < \frac{1}{|k|}$ .

Practice — then mark it

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

Question 1 of 2[3]

9709/12 · Q1

The coefficient of x^2 in the expansion of (1 - 4x)^6 is 12 times the coefficient of x^2 in the expansion of (2 + ax)^5. Find the value of the positive constant a.

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

Question 2 of 2[5]

9709/11 · Q2

The coefficient of $\frac{1}{x}$ in the expansion of $(kx + \frac{1}{x})^5 + (1 - \frac{2}{x})^8$ is 74. Find the value of the positive constant $k$ .

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.

Cambridge International A-Level Mathematics 9709 Syllabus↗

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 9709/12 · Q1 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

← PREVIOUS1.5 Trigonometry NEXT →1.7 Differentiation

Series

BY THE END, YOU CAN…

1Understand and use the binomial expansion of (1+x)ⁿ for any rational n and |x|<1.
2Recognise and work with arithmetic progressions, using formulae for the n-th term and the sum of n terms.
3Recognise and work with geometric progressions, using formulae for the n-th term and the sum of n terms.
4Understand convergence for a geometric series and use the formula for the sum to infinity.

In simple terms

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

Building with Patterns

THINK OF IT LIKE…

1
Binomial expansion: (1 + x)ⁿ for |x| < 1 when n is not a positive integer.
2
Arithmetic series: u_n = a + (n−1)d; sum S_n = n/2(2a + (n−1)d).
3
Geometric series: u_n = ar^(n−1); sum S_n = a(1−r^n)/(1−r) for r ≠ 1.
4
Sum to infinity exists when |r| < 1: S_∞ = a/(1−r).

Explore the concept

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

INTERACTIVESTEP 1 / 3 · Terms

Each term is the previous one times r: a, ar, ar², …

Key formulas

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

$u_n = a + (n-1)d$

Tap a symbol — great for exam definitions

$u_n = ar^{n-1}$

Tap a symbol — great for exam definitions

$S_\infty = \frac{a}{1-r}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Binomial Expansion for any Rational Index

For any rational $n$ , and for $|x|<1$ :

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$

This formula only works when the first term in the bracket is 1.
To expand $(a+bx)^n$ , you must first write it as $a^n(1+\frac{b}{a}x)^n$ .
The expansion is an infinite series, so questions will ask for the first few terms.
The validity condition is crucial: for $(1+kx)^n$ , the expansion is valid when $|kx|<1$ , or $|x| < \frac{1}{|k|}$ .

Arithmetic Progressions (AP)

n-th term: $u_n = a + (n-1)d$

Sum of first n terms: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a+l)$

Geometric Progressions (GP)

n-th term: $u_n = ar^{n-1}$

Sum of first n terms: $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$

Sum to Infinity of a Geometric Progression

Sum to infinity: $S_\infty = \frac{a}{1-r}$ , valid only for $|r|<1$ .

Worked examples

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

EXAMPLE 1

i) Find the first three terms in the expansion of $\sqrt{1-4x}$ in ascending powers of $x$ .

ii) State the range of values of $x$ for which the expansion is valid.

1
i) First, rewrite the expression in index form: $\sqrt{1-4x} = (1-4x)^{1/2}$ . This is in the form $(1+y)^n$ with $y = -4x$ and $n = \frac{1}{2}$ .

EXAMPLE 2

The second term of a geometric progression is 12 and its sum to infinity is 50. The common ratio is positive. Find the first term of the progression.

1
Let the first term be $a$ and the common ratio be $r$ .

How it all connects

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

MAIN IDEASeries

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 difference between a sequence and a series?

TAP OR SPACE TO FLIP

ANSWER

A sequence is a list of numbers in a specific order (e.g., 2, 4, 6, 8). A series is the sum of the terms of a sequence (e.g., 2 + 4 + 6 + 8).

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

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

✓
This formula only works when the first term in the bracket is 1.
✓
To expand $(a+bx)^n$ , you must first write it as $a^n(1+\frac{b}{a}x)^n$ .
✓
The expansion is an infinite series, so questions will ask for the first few terms.
✓
The validity condition is crucial: for $(1+kx)^n$ , the expansion is valid when $|kx|<1$ , or $|x| < \frac{1}{|k|}$ .

Practice — then mark it

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

Question 1 of 2[3]

9709/12 · Q1

The coefficient of x^2 in the expansion of (1 - 4x)^6 is 12 times the coefficient of x^2 in the expansion of (2 + ax)^5. Find the value of the positive constant a.

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

Question 2 of 2[5]

9709/11 · Q2

The coefficient of $\frac{1}{x}$ in the expansion of $(kx + \frac{1}{x})^5 + (1 - \frac{2}{x})^8$ is 74. Find the value of the positive constant $k$ .

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.

Cambridge International A-Level Mathematics 9709 Syllabus↗

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 9709/12 · Q1 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

← PREVIOUS1.5 Trigonometry NEXT →1.7 Differentiation

Revise Series for Cambridge 9709 (1.6)

In simple terms

Explore the concept

Key formulas

Full topic notes

Binomial Expansion for any Rational Index

Arithmetic Progressions (AP)

Geometric Progressions (GP)

Sum to Infinity of a Geometric Progression

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

9709/12 · Q1

9709/11 · Q2

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

Binomial Expansion for any Rational Index

Arithmetic Progressions (AP)

Geometric Progressions (GP)

Sum to Infinity of a Geometric Progression

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

9709/12 · Q1

9709/11 · Q2

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Series