Type a function, pick the center a and the order n, and get the Taylor polynomial with exact coefficients, every derivative value f⁽ᵏ⁾(a), the error at a point of your choice and a graph of f(x) against its approximation. Center 0 gives the Maclaurin series.
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Use x as the variable. Supports + − * / ^, sin cos tan sec csc cot, asin acos atan, sinh cosh tanh, exp, ln, log, sqrt, cbrt, abs, pi and e. 2x and 3sin(x) work too.
0 gives a Maclaurin series. pi/4 works.
Highest power, 0 to 40.
Shows f(x), Pn(x) and the approximation error.
Graph: f(x) and its Taylor polynomial
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How to use the Taylor Series Calculator
Type the function f(x), for example e^x, sin(x)/x or ln(1+x), or tap an example.
Choose the center a: 0 for a Maclaurin series, or any point such as 1 or pi/4.
Set the order n, the highest power of (x − a) to keep.
Press Calculate Taylor series to see the polynomial, the coefficient table and the graph.
Enter a value in Compare at x to see how close Pₙ(x) is to f(x), then copy the polynomial or download the coefficients.
The Taylor polynomial formula
For a function f that can be differentiated n times at x = a, the Taylor polynomial of order n is
The coefficient of (x − a)k is ck = f(k)(a)/k!, so if you know the coefficient you also know the k-th derivative: f(k)(a) = k! · ck. That is why the table above lists both. The polynomial agrees with f in value, slope, curvature and every derivative up to order n at the center, which is what makes it the best polynomial approximation near a.
The remainder (error) after n terms is given by Lagrange's form of Taylor's theorem: Rn(x) = f(n+1)(ξ)/(n+1)! · (x − a)n+1 for some ξ between a and x. If |f(n+1)| ≤ M on that interval, then |Rn(x)| ≤ M·|x − a|n+1/(n+1)!. For sin x and cos x, M = 1, so P7 of sin x at x = 1 is guaranteed to be within 1/8! ≈ 0.0000248.
Common Maclaurin series
f(x)
Maclaurin series
Converges for
eˣ
1 + x + x²/2! + x³/3! + x⁴/4! + …
all x
sin x
x − x³/3! + x⁵/5! − x⁷/7! + …
all x
cos x
1 − x²/2! + x⁴/4! − x⁶/6! + …
all x
1/(1 − x)
1 + x + x² + x³ + …
|x| < 1
ln(1 + x)
x − x²/2 + x³/3 − x⁴/4 + …
−1 < x ≤ 1
arctan x
x − x³/3 + x⁵/5 − x⁷/7 + …
|x| ≤ 1
(1 + x)r
1 + rx + r(r−1)/2! · x² + r(r−1)(r−2)/3! · x³ + …
|x| < 1
sinh x
x + x³/3! + x⁵/5! + …
all x
cosh x
1 + x²/2! + x⁴/4! + …
all x
tan x
x + x³/3 + 2x⁵/15 + 17x⁷/315 + …
|x| < π/2
Tips for good approximations
Center the series close to where you need it. To approximate ln(1.2), expand ln(x) about a = 1, not about a distant point.
Watch the radius of convergence. Outside it the polynomial diverges however many terms you add; the graph makes this easy to see for 1/(1 − x).
Odd and even functions have only odd or only even powers in their Maclaurin series (sin x is odd, cos x is even), so P7 and P8 of sin x are the same polynomial.
Substitution is a shortcut: the series of e−x² is the series of eᵗ with t = −x², which is why its coefficients are 1, −1, 1/2, −1/6, …
Frequently asked questions
What is a Taylor series?
A Taylor series writes a smooth function as an infinite sum of powers of (x − a): f(x) = Σ f⁽ᵏ⁾(a)/k! · (x − a)ᵏ. Keeping the terms up to power n gives the Taylor polynomial Pₙ(x), which matches f and its first n derivatives at x = a.
What is the difference between a Taylor and a Maclaurin series?
A Maclaurin series is simply a Taylor series centered at a = 0, so it uses powers of x instead of powers of (x − a). For example the Maclaurin series of eˣ is 1 + x + x²/2! + x³/3! + …
How does the calculator find the derivatives?
It uses power-series arithmetic (automatic differentiation): each operation in your function, such as multiplication, division, exp, ln or sin, is applied to truncated series instead of single numbers. This gives the derivatives at the center to full machine precision without numerical differencing, and the coefficients are then shown as exact fractions when they match one.
How accurate is the Taylor polynomial?
Near the center it is usually very accurate, and the error shrinks as n grows if x is inside the radius of convergence. By Taylor's theorem the error is R_n(x) = f⁽ⁿ⁺¹⁾(ξ)/(n+1)! · (x − a)ⁿ⁺¹ for some ξ between a and x. The calculator shows the actual error at the x you choose.
Why does the polynomial get worse far from the center?
A Taylor series only converges inside its radius of convergence, which is the distance from a to the nearest point where the function stops being analytic (in the complex plane). For 1/(1 − x) or ln(1 + x) about 0 the radius is 1, so beyond |x| = 1 adding more terms makes the approximation worse, not better. For eˣ, sin x and cos x the radius is infinite.
Why do I get an error for some centers?
A Taylor series exists only where the function is infinitely differentiable. ln(x) and √x at 0, 1/x at 0, |x| at 0 and tan(x) at π/2 have no Taylor series there, so the calculator explains the problem instead of returning wrong numbers. Move the center to a point where the function is smooth, for example ln(x) about a = 1.
Can it expand functions like sin(x)/x?
Yes. Removable singularities are handled automatically: sin(x)/x about 0 gives 1 − x²/6 + x⁴/120 − …, and (eˣ − 1)/x gives 1 + x/2 + x²/6 + …
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.