AHL 5.19—Maclaurin series: IB DP Math AA Notes

What is a Maclaurin Series?

A Maclaurin series is a way of representing a function as an infinite polynomial centred at $x = 0$ . The motivation is simple: polynomials are among the easiest functions to differentiate, integrate, and evaluate — so if we can express a complicated function like $sin x$ or $e^{x}$ as a polynomial, we gain a powerful computational tool.

Maclaurin Series

The Maclaurin series of a function $f (x)$ is the infinite polynomial expansion:
$f (x) = f (0) + f^{'} (0) x + \frac{f ^{''} ( 0 )}{2 !} x^{2} + \frac{f ^{'''} ( 0 )}{3 !} x^{3} + \dots = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( 0 )}{n !} x^{n}$
where $f^{(n)} (0)$ denotes the $n$ th derivative of $f$ evaluated at $x = 0$ .

The key requirement is that the polynomial $P (x)$ must agree with $f (x)$ at $x = 0$ in every derivative — meaning $P (0) = f (0)$ , $P^{'} (0) = f^{'} (0)$ , $P^{''} (0) = f^{''} (0)$ , and so on for all orders.

Analogy

Think of it like sculpting a function from clay. Each derivative you match at $x = 0$ is like pressing the clay into a finer and finer mould — the more derivatives you match, the closer your polynomial hugs the original function near $x = 0$ .

Deriving the Coefficients

Suppose we want to find a polynomial $P (x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$ that matches every derivative of $f (x)$ at $x = 0$ . We determine the coefficients $a_{n}$ one at a time by substituting $x = 0$ after each differentiation.

Step 1: Set $x = 0$ in $P (x)$ :
$P (0) = a_{0} ⟹ a_{0} = f (0)$

Step 2: Differentiate and set $x = 0$ :
$P^{'} (x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \dots ⟹ P^{'} (0) = a_{1} = f^{'} (0)$

Step 3: Differentiate again and set $x = 0$ :
$P^{''} (x) = 2 a_{2} + 6 a_{3} x + 12 a_{4} x^{2} + \dots ⟹ P^{''} (0) = 2 a_{2} = f^{''} (0) ⟹ a_{2} = \frac{f ^{''} ( 0 )}{2 !}$

Step 4: Differentiate once more and set $x = 0$ :
$P^{'''} (x) = 6 a_{3} + 24 a_{4} x + \dots ⟹ P^{'''} (0) = 6 a_{3} = f^{'''} (0) ⟹ a_{3} = \frac{f ^{'''} ( 0 )}{3 !}$

The pattern is clear. In general:
$a_{n} = \frac{f ^{(n)} ( 0 )}{n !}$

Substituting back gives the Maclaurin series formula:
$f (x) = \sum_{n = 0}^{\infty} \frac{f ^{(n)} ( 0 )}{n !} x^{n}$

Note

The denominator $n!$ arises naturally from repeated differentiation of $x^{n}$ . The $n$ th derivative of $x^{n}$ is $n!$ , which is what we divide by to isolate each coefficient.

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What is a Maclaurin Series?

Deriving the Coefficients

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