Maclaurin Series Calculator

The Maclaurin Series Calculator expands a common function — eˣ, sin(x), cos(x), sinh(x), cosh(x), ln(1+x), 1/(1−x), 1/(1+x), arctan(x), √(1+x), or arcsin(x) — into its Maclaurin series (a Taylor series centered at 0), with exact rational coefficients, the function's true radius of convergence, and an optional evaluation against the actual function value.

How to use it

1. Pick a function f(x) from the dropdown.
2. Choose how many terms to show (the first n nonzero powers of x).
3. Optionally enter a point x to compare the truncated series to the actual value of f there.
4. Read the Maclaurin Series for the expansion, the Coefficients table for each term's exact fraction, and the Radius of Convergence for where the series is actually valid.

Why exact fractions instead of decimals?

Every function on this list has a well-known coefficient formula — a factorial, a fixed sign pattern, or a generalized binomial coefficient — so every coefficient is a genuine rational number. Rounding it to a decimal (0.166667 instead of 1/6) hides that and accumulates error if you chain several terms by hand. This calculator computes each numerator and denominator as an exact (arbitrary-precision) integer and reduces it to lowest terms, so what you see is the coefficient, not an approximation of it.

Why radius of convergence matters

A Maclaurin series is only useful where it actually converges. 1/(1 − x) and 1/(1 + x) both have radius 1 and diverge at both endpoints; ln(1 + x) and √(1 + x) also have radius 1 but converge at one endpoint and not the other; arctan(x) and arcsin(x) converge at both endpoints; and eˣ, sin(x), cos(x), sinh(x), and cosh(x) converge everywhere. Plugging a point outside that radius into the truncated polynomial gives a number that looks plausible but has nothing to do with the actual function — the Radius of Convergence field, and the domain check on the evaluation point, exist to catch that.

Need a different center, or a function not on this list?

This calculator only covers the 11 functions above, each with a genuinely exact closed-form coefficient. For any other function, or a Taylor series about a center other than 0, use the Taylor Series Calculator, which symbolically differentiates any function you type in about any center a (setting a = 0 there gives the same Maclaurin series, just with floating-point rather than exact-fraction coefficients). To check whether a series you already have converges, see the Series Convergence Test Calculator; to sum a whole infinite series to its closed form instead of expanding a function, see the Infinite Series Sum Calculator.

Privacy

This calculator runs entirely in your browser. Your chosen function, term count, and evaluation point are never uploaded, logged, or stored.