What is the difference between Taylor and Maclaurin expansions?

A Taylor polynomial about a uses coefficients f^(k)(a)/k! multiplying (x-a)^k. A Maclaurin polynomial uses a = 0. Set Centre to 0 for Maclaurin, or enter another real numeric centre for Taylor.

How do degree and coefficient output work?

Order selects the polynomial degree from 0 through 10. The result lists a term for each power through that degree, including zero terms, and one further term. It does not automatically find a requested number of nonzero terms or an infinite-series formula.

Can it expand sin(x) or linearize a function?

Yes. sin(x) about 0 to degree 5 gives x-x^3/6+x^5/120. Degree 1 gives the local linearization f(a)+f'(a)*(x-a). Higher degrees add the corresponding derivative coefficients.

Does it calculate an interval or radius of convergence?

No. The output is a finite polynomial, with a numeric check near the centre. It does not find an interval or radius of convergence, an infinite sum, a limit, or a rigorous remainder bound. The next omitted term is not a bound on the total error.

Can it expand in two variables or solve differential equations?

Taylor mode expands in one variable at a time. Other symbols are held constant. It does not construct a joint multivariable expansion, work with complex centres, or solve differential equations using a series.

Taylor Series Calculator

Enter a function, one variable, a centre and a degree to build a finite Taylor polynomial. The opening Maclaurin example expands e^x about 0 to degree 3 as 1+x+x^2/2+x^3/6. Each included term and the next omitted term appear beside the result.

expressionorder 3 about x = 0

f = e^x variable       xcentre         x = 0order          3held constant  none

derivativesworking

Working out T₃f(x). Working out T₃f(x).

Mode

Expression

Variable

Order

Centre

T₃f(x): working
First term left out: working
Expansion: order 3 about x = 0
Result: Working

Working out T₃f(x).

∂/∂x of x²y + sin(y)y held constant
2*x*y
Then ∂/∂y of thatmixed second order, asked both ways round
2*x
x² + y² - 1 = 0, implicitdy/dx = -Fx/Fy, singular where 2*y = 0
-x/y

Accuracy. Symbolic differentiation in a sandboxed worker, exact, with the differentiation order printed beside the answer because the order you pick is part of the question. Every symbol you did not list is treated as a constant and the page says which those were. A function outside the supported set is named as unsupported rather than quietly dropped from the expression.

How this is worked out

The algebra is nerdamer 1.1.13, running in a worker of its own with a 2 second clock beside it: some expressions send it a long way, and the page stops it rather than freezing with it. Each step is a question of its own, nested inside the step before it, so the second partial is the derivative of the first and is checked against it rather than against the expression you typed.

Nothing the library returns is printed until it has been checked here on the page. Every derivative is compared against a slope measured on the expression that went into it, by central difference with one Richardson step, at up to 6 points, and they have to agree to within 0.0001 of the larger of one and the value. A mixed second partial is asked for a second time in the opposite order and the two answers are compared. An implicit result is checked against the curve itself: the page solves the relation for the dependent variable at a point, moves the independent one a little, solves again, and measures the slope between them. A Taylor expansion has to match the function at the centre and its error has to fall by the factor its order requires when the distance from the centre is halved. When a check contradicts the algebra, the page prints neither and shows what each one said.

What you can type

Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x and 3 sin(x) read the way they are written. Expressions are read up to 500 characters, a question may ask for up to 5 differentiations, and a Taylor expansion goes to order 10. A function outside that list is named as unsupported rather than dropped from the expression.

Common questions

What is the difference between Taylor and Maclaurin expansions?: A Taylor polynomial about a uses coefficients f^(k)(a)/k! multiplying (x-a)^k. A Maclaurin polynomial uses a = 0. Set Centre to 0 for Maclaurin, or enter another real numeric centre for Taylor.
How do degree and coefficient output work?: Order selects the polynomial degree from 0 through 10. The result lists a term for each power through that degree, including zero terms, and one further term. It does not automatically find a requested number of nonzero terms or an infinite-series formula.
Can it expand sin(x) or linearize a function?: Yes. sin(x) about 0 to degree 5 gives x-x^3/6+x^5/120. Degree 1 gives the local linearization f(a)+f'(a)*(x-a). Higher degrees add the corresponding derivative coefficients.
Does it calculate an interval or radius of convergence?: No. The output is a finite polynomial, with a numeric check near the centre. It does not find an interval or radius of convergence, an infinite sum, a limit, or a rigorous remainder bound. The next omitted term is not a bound on the total error.
Can it expand in two variables or solve differential equations?: Taylor mode expands in one variable at a time. Other symbols are held constant. It does not construct a joint multivariable expansion, work with complex centres, or solve differential equations using a series.

Symbolic differentiation in a sandboxed worker, exact, with the differentiation order printed beside the answer because the order you pick is part of the question. Every symbol you did not list is treated as a constant and the page says which those were. A function outside the supported set is named as unsupported rather than quietly dropped from the expression.