Taylor Expansion

Equivalent forms

f(x) = f(a) + f'(a)(x-a) + \tfrac{1}{2}f''(a)(x-a)^2 + \cdots

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

\sin x = x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \cdots

Every linearization in physics — small-angle pendulum, relativistic KE reducing to ½mv², weak-field gravity — is the first non-trivial term of this one series.

Symbol	Name	SI unit	Dimension	Range
$x$	Evaluation point Where the function is approximated	1	1	-6.283 – 6.283
$a$	Expansion center Point about which the series is built	1	1	-3 – 3
$N$	Order Highest power kept in the partial sum	1	1	1 – 12
$f(x)$	Function valueoutput Approximated value from the truncated series	1	1	-2 – 2

Unit systems

Symbol	SI	CGS	Imperial
$x$	1	1	1

Where it holds

Converges within the radius of convergence; for entire functions (e^x, sin, cos) it converges for all x, though slowly when far from a.

Dimensional analysis

f(x)

\Sigma f^{(n)}

(a)(x-a)^n/n!

Each term carries the dimension of f: f^{(n)}(a) has f's dimension divided by x^n, times (x-a)^n restores it. Homogeneous.

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Discovery

Brook Taylor · 1715

Taylor published the general series in his 1715 Methodus Incrementorum. The a = 0 special case is named for Colin Maclaurin, who popularised it. Physicists live in the first two terms — nearly every 'small-angle', 'weak-field', or 'low-velocity' approximation in physics is a truncated Taylor series.

Try this

Your calculator has no idea what sin(1.3) is — it only adds and multiplies. Taylor's series is the trick that turns any smooth function into an infinite polynomial a machine can actually compute.

Add one polynomial term at a time and watch the parabola-turned-snake climb toward sin(x), matching it perfectly near x = 0 and creeping outward with every order.

Research status: stable

Real-world applications

Calculator/CPU evaluation of sin, cos, exp, log
Small-angle approximation $\sin \theta \approx \theta for$ pendulums and optics
Perturbation theory in quantum mechanics and celestial mechanics
Linearizing nonlinear equations of motion around equilibrium

Common misconceptions

A Taylor series always converges to the function — it can diverge, or converge to the wrong value (e.g. $e^{-1/x^{2}}$ has an all-zero series at 0).
More terms are always better — for a fixed x, an asymptotic series can get worse past an optimal order.
Maclaurin and Taylor series are different tools — Maclaurin is just Taylor with $a = 0$ .

Experimental verification

Underlies every calculator/CPU transcendental function, the small-angle pendulum period, and the reduction of relativistic energy

\gamma mc^{2} to mc^{2} + \tfrac{1}{2}mv^{2} at low

speed — all experimentally confirmed to high precision.

Derivation

Assume

f(x) = \Sigma c_n (x

-a)^n.

Differentiate k times and

set x = a

: all terms vanish except the k-th, giving f^{(k)}(a

) = k! c_k

, hence

c_k = f^{(k)}

(a)/k!.

Substituting recovers the series.

Limiting cases

x \to a

⟶

f(x) \to f

(a)At the center every correction term vanishes; the series is exact.

N = 1

⟶

f(x) \approx f

(a) + f'(a)(x-a)The tangent-line (linear) approximation — the workhorse of physics.

|x - a| large⟶ truncation error growsFar from the center you need many more terms, or the series may diverge entirely.

What if…

What if the function isn't smooth at a?

The series doesn't exist there — Taylor needs all derivatives at the center. Use a Laurent or Fourier series instead.

What if I expand around a different point a?

You get a different-looking polynomial that still equals the same function, now most accurate near the new center.

1

Maclaurin series of e^x to 3rd order

Given ·

a:: 0
f:: e^x

Find · cubic approximation

Steps

All derivatives of e^x equal e^x, and $e^0 = 1$
$c_n = 1/n!$
Sum the first four terms

Result ·

e^x \approx 1 + x + x^{2}/2 + x^{3}/6

2

Why ½mv² is hidden in relativity

Given ·

E:: $\gamma mc^{2}$

Find · low-speed limit

Steps

$\gamma = (1 - v^{2}/c^{2})^{-1/2}$
Expand: $\gamma \approx 1 + \tfrac{1}{2} v^{2}/c^{2}$
Multiply by $mc^{2} to$ get rest energy + Newtonian KE

Result ·

E \approx mc^{2} + \tfrac{1}{2}mv^{2}

stirling approximationEuler's Formula→Fourier Series→taylor series complex