Taylor Series Calculator: Maclaurin Expansion with Steps

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Formula at a glance

Taylor Series Calculator: Taylor series — f(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ / k!. Example: Maclaurin series of eˣ → Taylor series: x⁵/120 + x⁴/24 + x³/6 + x²/2 + x + 1 + O(x⁶) — How the Taylor Series Calculator works: Taylor series — f(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ / k!

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📘 About the Taylor Series Calculator

A Taylor series writes a function as an infinite polynomial built from its derivatives at one point. Cut off after a few terms, it gives an excellent approximation near that point — which is how calculators compute sin, cos and eˣ, and how physicists make small-angle approximations like sin θ ≈ θ.

Choose the function, the centre and the order. The calculator differentiates repeatedly, evaluates each derivative at the centre (exactly where possible), divides by k! and assembles the polynomial. The graph compares the polynomial with the function, and you can check the error at any x.

🧮 Formula

Taylor series

f(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ / k!

Maclaurin (a = 0)

eˣ = 1 + x + x²/2! + x³/3! + …

sin and cos

sin x = x − x³/3! + x⁵/5! − …, cos x = 1 − x²/2! + x⁴/4! − …

✏️ Worked examples

Maclaurin series of eˣ

You enter
Function f(x)	e^x
Variable	x
Centre a	0
Order (highest power)	5

You get
Taylor series	x⁵/120 + x⁴/24 + x³/6 + x²/2 + x + 1 + O(x⁶)
Type	Maclaurin series (centred at 0), up to order 5
Accuracy at x = 0.5	series 1.648697917 vs exact 1.648721271 (error 0.0000234)

x⁵/120 + x⁴/24 + x³/6 + x²/2 + x + 1 + O(x⁶).

sin x to order 7

You enter
Function f(x)	sin(x)
Variable	x
Centre a	0
Order (highest power)	7

You get
Taylor series	−x⁷/5040 + x⁵/120 − x³/6 + x + O(x⁸)
Type	Maclaurin series (centred at 0), up to order 7
Accuracy at x = 0.5	series 0.4794255332 vs exact 0.4794255386 (error 5.37 × 10^-9)

−x⁷/5040 + x⁵/120 − x³/6 + x + O(x⁸).

🧭 How to use this calculator

What you enter

Function f(x)Use ^ for powers, * or nothing for ×: 3x^2 + 2sin(x), e^(2x), sqrt(x+1), ln(x).
VariableOne letter. Other letters are treated as constants.
Centre a
Order (highest power)
Check accuracy at x = (optional)

What you get

Taylor series
Type
Accuracy at x = 0.5

❓ Frequently asked questions

What is the difference between a Taylor and a Maclaurin series?

A Maclaurin series is a Taylor series centred at 0. Taylor series can be centred at any point a.

How many terms do I need?

More terms give a better approximation over a wider range. The error is roughly the size of the first term you leave out.

What is the Taylor series of ln(1 + x)?

x − x²/2 + x³/3 − x⁴/4 + …, valid for −1 < x ≤ 1.

≈ Taylor Series Calculator (Maclaurin Series)