Maclaurin Series Calculator

The most common mistake on this subject is believing that a Maclaurin series equals the function for every value of x. That is false. A Maclaurin series equals the function only inside its interval of convergence. Outside that interval, the series either diverges or converges to a different number entirely. This Maclaurin series calculator shows you the polynomial, the error, and a convergence warning when you try to use the series outside its valid range. Use it to see, term by term, whether adding more terms helps or makes things worse.

Open the calculator and select a function from the dropdown: e^x, sin(x), cos(x), ln(1+x), 1/(1-x), √(1+x), arctan(x), sinh(x), or cosh(x). Alternatively, enter a custom function using standard notation: ^ for power, * for multiplication, / for division, sin(), cos(), exp(), ln(), sqrt(). The custom function option uses numerical derivatives to construct the series, so it works for any differentiable function.

Set the number of nonzero terms. Choose a value between 1 and 20. The calculator will display that many nonzero terms in the expansion. For sin(x) and cos(x), nonzero terms are only the odd or even powers, respectively, so the polynomial degree will be higher than the number of terms.

Enter the x-value where you want to evaluate the approximation. The calculator computes the partial sum, the actual function value, the absolute error, and the relative error. If the x-value lies outside the interval of convergence, the calculator warns you that the partial sums do not approach the function.

Adjust the decimal places (2-8) and decide whether to show the term-by-term breakdown and the visualization chart. The chart plots the original function and the series approximation over the x-range you specify, so you can see where they match and where they diverge.

Click Calculate. The results appear in three sections: primary approximation, term table, and convergence information. Use Reset to clear all inputs and start over.

A Maclaurin series is a special case of the Taylor series centered at x = 0. The general formula is:

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... = Σ(n=0 to ∞) [f⁽ⁿ⁾(0)/n!] xⁿ

Where f⁽ⁿ⁾(0) is the nth derivative of f evaluated at x = 0, and n! is the factorial of n. The series represents the function as an infinite polynomial. The coefficients are determined entirely by the derivatives at the single point x = 0.

What the Formula Means

The term for n = 0 is f(0), which is just the function value at zero. The term for n = 1 is f'(0) x, which captures the slope at zero. Each higher term adds information from higher derivatives, weighted by 1/n! so that the series can converge. The factorial in the denominator grows faster than any power of x, which is why series for functions like e^x, sin x, and cos x converge for all real values of x.

The common Maclaurin series are fixed and appear in Stewart's 'Calculus' §11.10 Table 1 (9th edition, 2021, Cengage). They include e^x, sin x, cos x, ln(1+x), 1/(1-x), arctan x, (1+x)^k, sinh x, and cosh x. Knowing these by heart saves time because many other series can be derived from them through substitution, differentiation, or integration.

Take the function f(x) = e^x. Its Maclaurin series is Σ xⁿ/n! from n=0 to ∞. Every derivative of e^x is e^x, and at x=0 each derivative equals 1, so all coefficients are 1/n!.

Using the calculator, select e^x, set the number of nonzero terms to 5, and evaluate at x = 1. The terms are:

• n = 0: 1/0! = 1
• n = 1: 1/1! = 1
• n = 2: 1/2! = 0.5
• n = 3: 1/3! = 0.166667
• n = 4: 1/4! = 0.041667

The partial sum is 1 + 1 + 0.5 + 0.166667 + 0.041667 = 2.708333. The actual value of e¹ is 2.718282. The absolute error is 0.009949, and the relative error is 0.37%. Five terms give a good approximation at x = 1 because e^x converges for all x and the factorial denominator keeps the higher terms small.

Now evaluate at x = 3 using the same 5 terms. The terms become:

• n = 0: 1
• n = 1: 3
• n = 2: 9/2 = 4.5
• n = 3: 27/6 = 4.5
• n = 4: 81/24 = 3.375

Partial sum = 16.375. Actual e³ = 20.0855. Error = 3.7105, relative error = 18.5%. The approximation is poor because x is far from 0. The series still converges, but it needs many more terms to become accurate. The ratio test gives the radius of convergence as ∞, so the series will eventually converge, but the number of terms required grows with |x|.

A Maclaurin series is built from derivatives evaluated at x = 0. The higher the derivative order, the more information the series has about the function near zero. Far from zero, that information becomes less relevant. For functions with finite radius of convergence, such as ln(1+x) or 1/(1-x), the series simply does not converge outside that radius. For functions with infinite radius, like e^x or sin x, the series converges everywhere, but the number of terms needed for a given accuracy grows with |x|.

The calculator warns when the evaluation point lies outside the interval of convergence. For example, ln(1+x) converges only for -1 < x ≤ 1. At x = 2, the series for ln(1+x) diverges, and the partial sums do not approach ln(3). Adding more terms makes the error worse. The calculator flags this with a warning and reports the partial sum as 'Partial sum (series diverges here)'.

For functions with radius of convergence = 1, such as 1/(1-x), the series diverges for |x| ≥ 1. At x = 1, the function is undefined. At x = -1, the series 1 - 1 + 1 - 1 + ... oscillates and does not converge. The ratio test gives the radius, but endpoints must be checked separately. For 1/(1-x), both endpoints diverge. For ln(1+x), x = 1 gives the alternating harmonic series, which converges conditionally to ln(2). The calculator applies these endpoint checks.

Maclaurin series appear throughout physics, engineering, and numerical computation. In physics, perturbation theory replaces a nonlinear function with its Maclaurin series truncated to a few terms. The small-angle approximation sin θ ≈ θ is the first term of the Maclaurin series for sin x. For larger angles, including the θ³/6 term corrects for the nonlinearity. In classical mechanics, the series for (1+x)ᵏ approximates relativistic corrections.

In electrical engineering, the exponential current-voltage relationship of a diode is linearized using the Maclaurin series of exp(x) for small-signal analysis. The linear term gives the dynamic resistance, and higher terms model harmonic distortion in RF amplifiers. In digital signal processing, the series for sin x and cos x generate lookup tables for waveform synthesis on embedded systems with limited memory.

In numerical analysis, Maclaurin series provide polynomial approximations for functions that are expensive to compute directly. Calculators and computers use these series to evaluate sin, cos, exp, and ln. The number of terms is chosen so that the error falls below machine precision. The same idea underlies the CORDIC algorithm and other fast computation methods.

The series also appear in solving differential equations. When a closed-form solution is impossible, a power series solution can be found by assuming y(x) = Σ aₙ xⁿ and substituting into the equation. This is standard for equations with ordinary points, such as Bessel's equation or Legendre's equation.

The Lagrange remainder gives an exact expression for the error in a Maclaurin polynomial: Rₙ(x) = f⁽ⁿ⁺¹⁾(c) xⁿ⁺¹/(n+1)! for some c between 0 and x. Taylor's inequality turns this into a bound by replacing f⁽ⁿ⁺¹⁾(c) with its maximum on the interval. That bound is often 10 times larger than the actual error or more. It is a guarantee, not a prediction. The calculator reports the actual error, which is smaller, but for a theoretical proof you need the bound.

The one thing that most often goes wrong is forgetting to check whether x is inside the interval of convergence before using the series. The calculator catches that and warns you. Trust the warning.