Maclaurin and Taylor series (HL only)

IB Mathematics AA HL· Topic 5: Calculus, 5.14· 45 min read

1. Core Definitions★★☆☆☆HL only⏱ 15 min

📘 Definition

Taylor Series

For a function $f (x)$ that is infinitely differentiable at $x = a$ (called the center of expansion), the Taylor series is given by:

Example:

Expands $f (x)$ around any point $x = a$

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n

📘 Definition

Maclaurin Series

A special case of Taylor series where the center of expansion is $a = 0$ , given by:

Example:

Expands $f (x)$ around the origin

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n

📐 Worked Example

Find the first 3 non-zero terms of the Maclaurin series for $f (x) = e^{x}$

1
Calculate derivatives and evaluate at $a = 0$ :
2
$f (x) = e^{x} ⟹ f (0) = 1$ $f^{'} (x) = e^{x} ⟹ f^{'} (0) = 1$ $f^{''} (x) = e^{x} ⟹ f^{''} (0) = 1$
3
Substitute into the Maclaurin formula:
4
$n = 0$ term: $\frac{1}{0 !} x^{0} = 1$ $n = 1$ term: $\frac{1}{1 !} x^{1} = x$ $n = 2$ term: $\frac{1}{2 !} x^{2} = \frac{x ^{2}}{2}$
5
Final result (first 3 non-zero terms):
6
$e^x \approx 1 + x + \frac{x^2}{2}$

Exam tip:

You can be asked to derive the general formula for a Taylor or Maclaurin series from first principles, so memorize the general form.

2. Standard Maclaurin Series★★★☆☆HL only⏱ 25 min

There are 6 standard Maclaurin series you should memorize for exams, along with their intervals of convergence:

Function	Maclaurin Series	Interval of Convergence
$e^{x}$	$\sum_{n = 0}^{\infty} \frac{x ^{n}}{n !}$	$(- \infty, \infty)$
$sin x$	$\sum_{n = 0}^{\infty} \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!}$	$(- \infty, \infty)$
$cos x$	$\sum_{n = 0}^{\infty} \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!}$	$(- \infty, \infty)$
$ln (1 + x)$	$\sum_{n = 1}^{\infty} \frac{( - 1 ) ^{n + 1} x ^{n}}{n}$	$(- 1, 1]$
$(1 + x)^{k}$	$\sum_{n = 0}^{\infty} (n k) x^{n}$	$(- 1, 1)$
$\frac{1}{1 - x}$	$\sum_{n = 0}^{\infty} x^{n}$	$(- 1, 1)$

📐 Worked Example

Use the standard $cos x$ series to find the first 4 non-zero terms of the Maclaurin series for $cos (2 x)$

1
Recall the standard expansion for $cos u$ :
2
$\cos u = 1 - \frac{u^2}{2!} + \frac{u^4}{4!} - \frac{u^6}{6!} + ...$
3
Substitute $u = 2 x$ :
4
$\cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \frac{(2x)^6}{6!} + ...$
5
Simplify each coefficient:
6
$1 - \frac{4x^2}{2} + \frac{16x^4}{24} - \frac{64x^6}{720} = 1 - 2x^2 + \frac{2x^4}{3} - \frac{8x^6}{45} + ...$

✓ Quick check

What is the interval of convergence for the Maclaurin series of $ln (1 + x)$ ?
- $(- \infty, \infty)$
- $[- 1, 1]$
- $(- 1, 1]$
- $(- 1, 1)$
Reveal answer
2 —
The series $\sum \frac{( - 1 ) ^{n + 1} x ^{n}}{n}$ converges when $- 1 < x \leq 1$ , diverges at $x = - 1$ .

Exam tip:

Memorizing standard series saves 5-10 minutes per question compared to deriving them from scratch in an exam.

3. Taylor Series About Non-Zero Centers★★★★☆HL only⏱ 25 min

To expand a function around any center $a \neq = 0$ , we use the general Taylor series formula. A useful shortcut is substitution: let $u = x - a$ , expand $f (a + u)$ as a Maclaurin series in $u$ , then substitute back $u = x - a$ .

📐 Worked Example

Find the first 3 non-zero terms of the Taylor series for $f (x) = ln x$ centered at $a = 1$ .

1
Calculate derivatives and evaluate at $a = 1$ :
2
$f (x) = ln x ⟹ f (1) = 0$ $f^{'} (x) = \frac{1}{x} ⟹ f^{'} (1) = 1$ $f^{''} (x) = - \frac{1}{x ^{2}} ⟹ f^{''} (1) = - 1$ $f^{'''} (x) = \frac{2}{x ^{3}} ⟹ f^{'''} (1) = 2$
3
Substitute into the Taylor formula:
4
$f(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3 + ...$
5
Simplify to get the first 3 non-zero terms:
6
$\ln x \approx (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3}$

4. Applications: Approximation and Limits★★★★☆HL only⏱ 20 min

Two of the most common exam applications of Maclaurin/Taylor series are approximating function values near the center, and evaluating indeterminate limits that would be cumbersome with L'Hospital's rule.

📐 Worked Example

Evaluate $lim_{x \to 0} \frac{e ^{x} - 1 - x}{x ^{2}}$ using Maclaurin series.

1
Substitute the Maclaurin expansion of $e^{x}$ into the numerator:
2
$e^x - 1 - x = \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...\right) - 1 - x = \frac{x^2}{2} + \frac{x^3}{6} + ...$
3
Divide the entire expression by $x^{2}$ :
4
$\frac{e^x - 1 - x}{x^2} = \frac{1}{2} + \frac{x}{6} + \frac{x^2}{24} + ...$
5
Take the limit as $x \to 0$ : all terms containing $x$ approach 0, so the limit is:
6
$\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \frac{1}{2}$

✓ Quick check

What is the center of the Taylor series: $\sum_{n = 0}^{\infty} \frac{( - 1 ) ^{n} ( x - 2 ) ^{n}}{n !}$
- 0
- 2
- -2
- n
Reveal answer
1 —
The general term of a Taylor series is $\frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$ , so $a = 2$ here.

5. Common Pitfalls

Wrong move:

Forgetting the $n!$ denominator in series terms

Why:

This is the most common avoidable mistake, and examiners always penalize missing factorials

Correct move:

Always write out the general formula with $n!$ before substituting derivative values

Wrong move:

Trying to find a Maclaurin series for a function undefined at 0, e.g. $ln x$

Why:

All derivatives will be undefined at 0, so no valid expansion can be found

Correct move:

Use Taylor series centered at a point where the function is defined, e.g. $a = 1$ for $ln x$

Wrong move:

Forgetting to raise the substitution constant to the power of the term, e.g. writing $(k x)^{n} = k x^{n}$

Why:

This leads to incorrect coefficients for all terms after the first

Correct move:

Simplify $(k x)^{n} = k^{n} x^{n}$ for every term when substituting into a standard series

Wrong move:

Using a series outside its interval of convergence

Why:

The series does not equal the function outside this interval, so results are meaningless

Correct move:

Always confirm your $x$ value lies within the interval of convergence before using the series

Wrong move:

Treating the binomial series for non-integer $k$ as a finite expansion

Why:

Only for positive integer $k$ is the binomial expansion finite; for all other $k$ , it is infinite

Correct move:

Use the infinite binomial series with interval of convergence $(- 1, 1)$ for non-integer $k$

6. Quick Reference Cheatsheet

Type	Formula	Key Note
Maclaurin (center 0)	$\sum_{n = 0}^{\infty} \frac{f ^{(n)} ( 0 )}{n !} x^{n}$	Special case of Taylor
Taylor (center $a$ )	$\sum_{n = 0}^{\infty} \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}$	Works for any center $a$
$e^{x}$	$\sum \frac{x ^{n}}{n !}$	All real $x$
$sin x$	$\sum \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!}$	All real $x$
$cos x$	$\sum \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!}$	All real $x$
$ln (1 + x)$	$\sum \frac{( - 1 ) ^{n + 1} x ^{n}}{n}$	$- 1 < x \leq 1$
$(1 + x)^{k}$	$\sum (n k) x^{n}$	$- 1 < x < 1$ , all $k$

7. Frequently Asked

When do I use Taylor series instead of Maclaurin series?

Use Maclaurin only when you are expanding around $a = 0$ . Use Taylor when expanding around any other center, or when the function is undefined at $a = 0$ (e.g. $ln x$ has no Maclaurin series).

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 2
Taylor series approximation
2022 · 1
Limit via Maclaurin series
2023 · 2
Maclaurin series derivation

What's Next

Maclaurin and Taylor series are foundational tools for advanced calculus, used in error estimation, solving differential equations, and numerical analysis. In IB AA HL, the next core topic related to this sub-topic is Lagrange error bounds, which teaches you to calculate the maximum error of a Taylor polynomial approximation. Understanding series expansions also makes evaluating complex indeterminate limits much easier, and is used to solve many types of differential equations that cannot be solved with standard algebraic methods.

→Fundamental theorem of calculus