Taylor Series #

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Infinite polynomial approximation of functions around a point.

Interactive Visualization #

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t=0s

Core Concepts #

-Power-series representation: a function expressed (when possible) as an infinite sum of powers about a point a, sum_{n>=0} c_n (x-a)^n
-Derivative-based coefficients: the coefficient of (x-a)^n is given by the nth derivative at the center divided by n!, i.e. c_n = f^{(n)}(a)/n!
-Nth-degree Taylor polynomial: the finite truncation of the power series up to n gives the polynomial approximation of degree n
-Radius of convergence: a (possibly zero or infinite) interval/region around a where the power series converges and may equal the function

Key Symbols & Notation #

Summation (sigma) notation for infinite series: "sum_{n=0}^infty"

Essential Relationships #

-Truncation/convergence link: the nth Taylor polynomial equals the truncated series using the derivative-based coefficients, and the remainder (f(x) minus that truncation) tends to zero on points where the full Taylor series converges to f

Prerequisites (2) #

Derivative Rules5 atoms Sequences5 atoms

Advanced Learning Details

Graph Position #

Depth Cost

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Bottleneck Score

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Cognitive Load #

Atomic Elements

Total Elements

Percentile Level

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All Concepts (13) #

- Taylor series: representing a function as an infinite power series centered at a point a
- Maclaurin series: Taylor series specialized to center a = 0
- Taylor polynomial: the finite partial sum (degree-n polynomial) used as an approximation to the function
- Power series: an infinite series of powers of (x - a) (family that includes Taylor series)
- Center of expansion: the point a about which the series is formed
- Remainder (error) term: the difference between the function and its Taylor polynomial
- Lagrange form of the remainder (specific formula expressing an error using an (n+1)th derivative at some intermediate point)
- Radius of convergence: the distance from the center within which the power series converges
- Interval of convergence: the set of x values for which the series converges (including endpoint behavior)
- Convergence vs. pointwise equality: distinction between a series converging and the series actually equaling the function
- Analytic function: a function that equals its Taylor series on some neighbourhood of the center
- Term-by-term differentiation and integration of power series (valid within radius of convergence)
- Order of approximation: how the error scales with (x-a) and polynomial degree (e.g., next unused power determines leading error term)

Teaching Strategy #

Deep-dive lesson - accessible entry point but dense material. Use worked examples and spaced repetition.

Many functions are complicated globally but simple locally. Taylor series formalize that idea: near a point a, a smooth function behaves like a polynomial whose coefficients are determined entirely by derivatives at a.

TL;DR:

A Taylor series expresses (when possible) a function f(x) as an infinite power series around a center a: f(x) = ∑_{n=0}^∞ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. Truncating after n terms gives the nth-degree Taylor polynomial, a practical approximation near a. The series may converge only within a radius of convergence R, and even where it converges it may or may not equal the original function—so you must check convergence and (when needed) remainder/error behavior.

What Is Taylor Series? #

Why we want it (motivation) #

Polynomials are the “friendly” functions of calculus:

•They’re easy to differentiate and integrate.
•They’re easy to compute numerically.
•They behave predictably near a point.

But many important functions are not polynomials: eˣ, sin x, cos x, ln x, 1/(1−x), etc. Taylor series is the bridge: it turns a function (near some center a) into an infinite polynomial-like object.

The core idea is local approximation: if you zoom in near x = a, a smooth function looks more and more like its tangent line; if you zoom in further, a quadratic improves the fit; then a cubic, and so on.

Definition (power-series representation) #

A power series about a is an infinite sum

∑_{n=0}^∞ cₙ (x−a)ⁿ.

A Taylor series is a particular power series for a function f whose coefficients are chosen to match derivatives of f at the center a:

f(x) = ∑_{n=0}^∞ [ f⁽ⁿ⁾(a) / n! ] (x−a)ⁿ.

The notation f⁽ⁿ⁾ means the nth derivative: f⁽⁰⁾ = f, f⁽¹⁾ = f′, f⁽²⁾ = f″, etc.

When a = 0, this special case is called a Maclaurin series:

f(x) = ∑_{n=0}^∞ [ f⁽ⁿ⁾(0) / n! ] xⁿ.

The “matching derivatives” intuition #

A polynomial of degree n can match up to n derivatives at a point. Taylor series pushes this to “match all derivatives,” term by term.

Consider a polynomial approximation Pₙ(x) of degree n around a:

Pₙ(x) = c₀ + c₁(x−a) + c₂(x−a)² + ⋯ + cₙ(x−a)ⁿ.

If we require that Pₙ and f have the same derivatives at a up to order n:

•Pₙ(a) = f(a)
•Pₙ′(a) = f′(a)
•…
•Pₙ⁽ⁿ⁾(a) = f⁽ⁿ⁾(a)

then the coefficients are forced to be

cₖ = f⁽ᵏ⁾(a)/k!.

This is the fundamental mechanism: derivatives at the center determine the coefficients.

Important subtlety #

A Taylor series can exist (all derivatives exist) but still:

converge only on a limited interval, and/or
converge but not equal the original function.

So the full story has two parts:

•Construct the series from derivatives.
•Analyze where it converges and whether it equals f.

Core mechanic 1: Derivative-based coefficients and the nth-degree Taylor polynomial #

Why truncation matters #

An infinite series is a theoretical object; in computation and estimation, we nearly always use a finite truncation.

The nth-degree Taylor polynomial of f about a is

Tₙ(x) = ∑_{k=0}^n [ f⁽ᵏ⁾(a) / k! ] (x−a)ᵏ.

This is the best polynomial of degree ≤ n that matches derivatives up to order n at x = a.

How the coefficients appear (showing the mechanism) #

Start with the general polynomial form around a:

Tₙ(x) = c₀ + c₁(x−a) + c₂(x−a)² + ⋯ + cₙ(x−a)ⁿ.

Differentiate term-by-term:

Tₙ′(x) = c₁ + 2c₂(x−a) + 3c₃(x−a)² + ⋯ + n cₙ(x−a)ⁿ⁻¹.

Evaluate at x = a (so every (x−a) term becomes 0):

Tₙ′(a) = c₁.

Differentiate again:

Tₙ″(x) = 2c₂ + 3·2 c₃(x−a) + 4·3 c₄(x−a)² + ⋯

Evaluate at x = a:

Tₙ″(a) = 2c₂ ⇒ c₂ = Tₙ″(a)/2!.

Continuing this pattern, the kth derivative at a isolates cₖ multiplied by k!:

Tₙ⁽ᵏ⁾(a) = k! cₖ ⇒ cₖ = Tₙ⁽ᵏ⁾(a)/k!.

If we impose Tₙ⁽ᵏ⁾(a) = f⁽ᵏ⁾(a) for k = 0,…,n, then

cₖ = f⁽ᵏ⁾(a)/k!.

This is where the factorial comes from: repeated differentiation pulls down k·(k−1)·…·1 = k!.

A practical workflow #

To build Tₙ(x) about a:

Compute f(a), f′(a), f″(a), …, f⁽ⁿ⁾(a).
Plug into

Tₙ(x) = f(a) + f′(a)(x−a) + f″(a)/2! (x−a)² + ⋯ + f⁽ⁿ⁾(a)/n! (x−a)ⁿ.

Quick comparison table: “Taylor polynomial vs Taylor series” #

Object	Notation	What it is	Used for	Caveat
Taylor polynomial	Tₙ(x)	Finite sum up to degree n	Approximate f near a	Has error (remainder)
Taylor series	∑_{n=0}^∞ …	Infinite sum	Exact representation when it converges to f	May converge only for

The remainder (error) idea #

Define the remainder after degree n as

Rₙ(x) = f(x) − Tₙ(x).

Taylor’s theorem (in one common form) says that if f has n+1 derivatives near a, then

Rₙ(x) = f⁽ⁿ⁺¹⁾(ξ) / (n+1)! · (x−a)ⁿ⁺¹

for some ξ between a and x.

Even if you don’t use this exact form yet, it communicates a key lesson:

•The error typically scales like (x−a)ⁿ⁺¹.
•Higher degree gives rapidly improved accuracy when |x−a| is small.

This explains why Taylor approximations are “local”: the small parameter is (x−a).

Core mechanic 2: Radius of convergence and when the series equals the function #

Why convergence is the gatekeeper #

A Taylor series is an infinite sum. Infinite sums are only meaningful if they converge.

For a power series

∑_{n=0}^∞ cₙ (x−a)ⁿ,

typically there exists a number R (0 ≤ R ≤ ∞) called the radius of convergence such that:

•The series converges for |x−a| < R.
•The series diverges for |x−a| > R.
•At |x−a| = R, convergence may or may not happen (must be checked separately).

So the “region where the series makes sense” is an interval (a−R, a+R) on the real line.

How we often find R (ratio test intuition) #

In many calculus settings, R is found via the ratio test. Consider terms

uₙ(x) = cₙ (x−a)ⁿ.

lim_{n→∞} |uₙ₊₁(x) / uₙ(x)|

= lim_{n→∞} |cₙ₊₁/cₙ| · |x−a|

= L · |x−a|,

then convergence typically requires L · |x−a| < 1, meaning |x−a| < 1/L. That value is R.

You don’t need every detail of series tests to use Taylor series effectively, but you do need the mindset:

•A Taylor series is not automatically valid “for all x.”
•Its validity is local to the center a and controlled by R.

Converging vs representing the function #

Even within |x−a| < R, the series sum might not equal f(x) unless f is “nice enough.”

A common sufficient condition (not the only one) is: if the remainder Rₙ(x) → 0 as n → ∞ for x in some interval, then

f(x) = lim_{n→∞} Tₙ(x) = ∑_{n=0}^∞ f⁽ⁿ⁾(a)/n! (x−a)ⁿ.

Many standard functions (eˣ, sin x, cos x, ln(1+x) on its interval, rational functions away from poles) behave well.

Singularities and why they control R (big picture) #

A powerful intuition: the radius of convergence is often limited by the nearest point where the function “breaks” (e.g., division by zero or non-analytic behavior).

Example intuition:

•f(x) = 1/(1−x) centered at 0 has a problem at x = 1, so R = 1.
•f(x) = 1/(1+x²) centered at 0 has complex singularities at x = ±i, distance 1 from 0 in the complex plane, so R = 1 (even though on the real line the function is perfectly finite everywhere).

You don’t have to master complex analysis here, but this perspective prevents surprises: the convergence radius is about the function’s analytic obstacles, not just real-valued smoothness.

Endpoint checking (common pattern) #

If R is finite, you must check x = a ± R separately. It’s common to see:

•Converges on one endpoint but not the other.
•Converges on both.
•Converges on neither.

That endpoint behavior matters when using a series to represent a function on a closed interval.

Application/Connection: Approximations, computation, and modeling #

Why Taylor series is a workhorse #

Taylor series is not just a calculus curiosity. It’s a central tool for:

•Approximating hard functions with polynomials.
•Estimating errors and understanding sensitivity.
•Deriving algorithms (numerical methods, scientific computing).
•Modeling near an operating point (physics, engineering, economics).

Local linearization and beyond #

You may already know the tangent-line approximation:

f(x) ≈ f(a) + f′(a)(x−a).

That is exactly T₁(x). Taylor series generalizes this:

•T₂ adds curvature (second derivative).
•T₃ adds asymmetry/inflection behavior.
•Higher orders capture more local geometry.

Even in multivariable calculus, a closely related idea appears (Taylor expansion with gradients and Hessians). You’ll later see vectors like x and a, and approximations using ∇f and the Hessian matrix. (In this lesson, we stay 1D, but the conceptual jump is small.)

Typical “standard expansions” you reuse constantly #

Certain Maclaurin series appear everywhere:

Exponential:

eˣ = ∑_{n=0}^∞ xⁿ/n! = 1 + x + x²/2! + x³/3! + ⋯

Sine and cosine:

sin x = ∑_{n=0}^∞ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!

cos x = ∑_{n=0}^∞ (−1)ⁿ x²ⁿ/(2n)!

Geometric series (a gateway to many others):

1/(1−x) = ∑_{n=0}^∞ xⁿ for |x| < 1.

From (3), many manipulations become possible: integrate term-by-term, differentiate term-by-term, substitute x → −x, etc.

Numerical computation mindset #

Suppose you need sin(0.1). A calculator uses algorithms that reduce to polynomial-like approximations internally.

Using Taylor:

sin x ≈ x − x³/3! + x⁵/5!.

At x = 0.1, higher powers shrink rapidly:

•0.1³ = 0.001
•0.1⁵ = 0.00001

So a few terms give high accuracy.

Modeling: choosing the center a #

The center a is not arbitrary—it’s a design decision.

•If you care about accuracy near x = 2, expand around a = 2.
•If you care near 0, use Maclaurin.

A good mental model:

•Taylor polynomial is like a custom-fit polynomial built for a neighborhood around a.

Connection to differential equations and optimization #

Later you’ll see:

•Solving ODEs via power series (assume y = ∑ cₙ(x−a)ⁿ and solve for coefficients).
•Newton’s method analysis uses Taylor expansion of f near a root.
•Optimization uses second-order Taylor (quadratic models) to approximate objectives.

Taylor series is one of the main reasons derivatives are so valuable: derivatives are not just slopes—they are information that determines local function behavior to arbitrary order.

Worked Examples (3) #

Build a Taylor polynomial for eˣ around a = 0 (Maclaurin) and approximate e^0.2 #

We want T₄(x) for f(x) = eˣ at a = 0, then use it to approximate e^0.2.

Compute derivatives:
f(x) = eˣ
f′(x) = eˣ
f″(x) = eˣ
f‴(x) = eˣ
f⁽⁴⁾(x) = eˣ
Evaluate at a = 0:
f(0) = 1
f′(0) = 1
f″(0) = 1
f‴(0) = 1
f⁽⁴⁾(0) = 1
Form the Taylor polynomial:
T₄(x) = ∑_{k=0}^4 f⁽ᵏ⁾(0)/k! · xᵏ
= 1 + x + x²/2! + x³/3! + x⁴/4!
Plug in x = 0.2:
T₄(0.2) = 1 + 0.2 + 0.2²/2 + 0.2³/6 + 0.2⁴/24
= 1 + 0.2 + 0.04/2 + 0.008/6 + 0.0016/24
= 1 + 0.2 + 0.02 + 0.001333… + 0.0000666…
≈ 1.2214666…
Compare intuition:
The true value is e^0.2 ≈ 1.221402…
The approximation is already accurate to about 4 decimal places with only 5 terms.

Insight: Because eˣ has derivatives that stay the same and factorials grow fast, the terms xⁿ/n! shrink quickly for modest |x|. That’s why truncations of eˣ are especially effective.

Derive the Maclaurin series for sin x and build a 5th-degree approximation #

We will compute derivatives of sin x at 0, identify the pattern, then write T₅(x) and use it as an approximation near 0.

Start with f(x) = sin x and compute derivatives:
f(x) = sin x
f′(x) = cos x
f″(x) = −sin x
f‴(x) = −cos x
f⁽⁴⁾(x) = sin x
f⁽⁵⁾(x) = cos x
Evaluate at 0:
f(0) = sin 0 = 0
f′(0) = cos 0 = 1
f″(0) = −sin 0 = 0
f‴(0) = −cos 0 = −1
f⁽⁴⁾(0) = sin 0 = 0
f⁽⁵⁾(0) = cos 0 = 1
Write the Taylor polynomial through degree 5:
T₅(x) = f(0) + f′(0)x + f″(0)/2! x² + f‴(0)/3! x³ + f⁽⁴⁾(0)/4! x⁴ + f⁽⁵⁾(0)/5! x⁵
Substitute the values:
T₅(x) = 0 + 1·x + 0·x² + (−1)/3! x³ + 0·x⁴ + 1/5! x⁵
= x − x³/6 + x⁵/120
Interpretation near 0:
sin x ≈ x − x³/6 + x⁵/120
The approximation improves as x gets closer to 0 because higher powers shrink fast.

Insight: Only odd powers appear because sin x is an odd function, and the derivatives at 0 alternate between 0, ±1. Symmetry of the function shows up directly in which Taylor coefficients vanish.

Find the radius of convergence for the series of 1/(1−x) and connect it to the function’s singularity #

Consider the geometric series ∑_{n=0}^∞ xⁿ. We’ll see when it converges and why it equals 1/(1−x) there.

Consider partial sums S_N = 1 + x + x² + ⋯ + xᴺ.
Multiply by (1−x):
(1−x)S_N = S_N − xS_N
= (1 + x + x² + ⋯ + xᴺ) − (x + x² + ⋯ + xᴺ + xᴺ⁺¹)
= 1 − xᴺ⁺¹.
So for x ≠ 1:
S_N = (1 − xᴺ⁺¹)/(1−x).
Now take N → ∞. If |x| < 1 then xᴺ⁺¹ → 0, so
lim_{N→∞} S_N = 1/(1−x).
If |x| > 1 then xᴺ⁺¹ does not go to 0, so the series diverges.
At |x| = 1, check endpoints:
x = 1 gives 1 + 1 + 1 + ⋯ diverges.
x = −1 gives 1 − 1 + 1 − 1 + ⋯ does not converge in the usual sense.
Therefore the radius of convergence is R = 1, and the series equals 1/(1−x) for |x| < 1.

Insight: The function 1/(1−x) has a singularity (division by zero) at x = 1, exactly one unit away from the center 0. That nearest breakdown point matches the radius of convergence R = 1.

Key Takeaways #

✓
A Taylor series is a power series centered at a: f(x) = ∑_{n=0}^∞ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ (when it converges to f).
✓
The nth-degree Taylor polynomial Tₙ(x) is the truncation up to n; it matches f and its first n derivatives at x = a.
✓
Factorials arise because the kth derivative of (x−a)ᵏ at a equals k!; this forces the coefficient cₖ = f⁽ᵏ⁾(a)/k!.
✓
Taylor approximations are local: accuracy is typically good when |x−a| is small, and improves with higher degree.
✓
Power series have a radius of convergence R: they converge for |x−a| < R and diverge for |x−a| > R; endpoints require separate checks.
✓
Convergence of the Taylor series is not the same as equality to f; to claim representation, you need the remainder Rₙ(x) → 0 (or other analytic guarantees).
✓
Standard expansions (eˣ, sin x, cos x, 1/(1−x), ln(1+x)) are reusable building blocks across calculus and applied math.

Common Mistakes #

✗
Assuming the Taylor series equals the function for all x without checking the radius of convergence (and endpoints).
✗
Forgetting the center a and incorrectly writing powers of x instead of (x−a) when expanding around a ≠ 0.
✗
Dropping the factorial: coefficients are f⁽ⁿ⁾(a)/n!, not just f⁽ⁿ⁾(a).
✗
Using a low-degree polynomial far from a and expecting good accuracy (Taylor is local, not global).

Practice #

medium

Compute the 3rd-degree Taylor polynomial T₃(x) for f(x) = ln(1+x) centered at a = 0.

Hint: Differentiate ln(1+x) repeatedly and evaluate at x = 0. Watch the alternating signs and factorials.

Show solution

f(x) = ln(1+x)

Derivatives:

f′(x) = 1/(1+x)

f″(x) = −1/(1+x)²

f‴(x) = 2/(1+x)³

Evaluate at 0:

f(0) = 0

f′(0) = 1

f″(0) = −1

f‴(0) = 2

Taylor polynomial:

T₃(x) = f(0) + f′(0)x + f″(0)/2! x² + f‴(0)/3! x³

= 0 + x + (−1)/2 x² + 2/6 x³

= x − x²/2 + x³/3.

medium

Find the radius of convergence of the power series ∑_{n=1}^∞ n(x−2)ⁿ.

Hint: Use the ratio test on uₙ = n(x−2)ⁿ. Simplify |uₙ₊₁/uₙ|.

Show solution

Let uₙ = n(x−2)ⁿ.

Compute the ratio:

= |(n+1)/n| · |x−2|

= (1 + 1/n) |x−2|.

Take n → ∞:

lim_{n→∞} |uₙ₊₁/uₙ| = 1 · |x−2| = |x−2|.

Ratio test gives convergence when |x−2| < 1.

So the radius of convergence is R = 1 (center a = 2).

easy

Use the Maclaurin polynomial for sin x up to x⁵ to approximate sin(0.3). Give the numerical value of x − x³/6 + x⁵/120 at x = 0.3.

Hint: Compute 0.3³ and 0.3⁵, then apply the coefficients 1/6 and 1/120.

Show solution

Use T₅(x) = x − x³/6 + x⁵/120.

At x = 0.3:

0.3³ = 0.027

0.3⁵ = 0.3²·0.3³ = 0.09·0.027 = 0.00243

Compute:

T₅(0.3) = 0.3 − 0.027/6 + 0.00243/120

= 0.3 − 0.0045 + 0.00002025

= 0.29552025.

So sin(0.3) ≈ 0.29552025 using the 5th-degree Maclaurin approximation.

Connections #

Power Series Basics

Convergence Tests

Maclaurin Series (Taylor at 0)

Newton’s Method

Multivariable Taylor Approximation

Error Bounds and Taylor’s Theorem

Quality: A (4.3/5)

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