STEM Calculators

Maclaurin Series Builder

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published Updated
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9. 9. Maclaurin Series Builder
Builds known Maclaurin series for common functions (sin, cos, \(e^x\), \(\ln(1+x)\)) up to a chosen order, optionally for composed variants \(A\,f(kx)\). Includes a zoomable, pannable plot of \(f(x)\) vs. its Maclaurin polynomial \(P_n(x)\).
Inputs
Uses known derivative patterns at 0 to build \(P_n(x)=\sum_{m=0}^{n}\frac{f^{(m)}(0)}{m!}x^m\).
Build for \(F(x)=A\,f(kx)\).
Example: \(A=1,\ k=3\) gives \(\cos(3x)\) or \(\sin(3x)\).
Polynomial includes powers up to \(x^n\). The remainder is reported as \(O(x^{n+1})\) if enabled.
Click a preset to load & compute.
Ready
Graph
Drag to pan • wheel/pinch to zoom. Plots \(F(x)=A\,f(kx)\) and its Maclaurin polynomial \(P_n(x)\).
More samples = smoother curves.
x: 0, y: 0, zoom(px/unit): 60
Results & steps
Click Fill example or press Calculate.

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Frequently Asked Questions

What is a Maclaurin series and how is it different from a Taylor series?

A Maclaurin series is a Taylor series centered at 0. It uses derivatives evaluated at x=0: f(x)=sum f^(m)(0)/m! x^m.

What does the Maclaurin polynomial P_n(x) represent?

P_n(x) is the finite truncation of the Maclaurin series up to degree n. It approximates the function near x=0 using the first n+1 derivative values at 0.

How does the builder handle F(x)=A f(kx)?

It applies the amplitude A and horizontal scaling k inside the chosen function and then constructs the Maclaurin polynomial for the resulting F(x). This changes the coefficients compared with the base f(x).

What does O(x^(n+1)) mean in the output?

It indicates the remainder order after truncating at degree n. Near x=0, the difference f(x)-P_n(x) behaves like a constant times x^(n+1) or smaller in magnitude.

Why would I enable prefer exact fractions?

Exact fractions avoid rounding in coefficients and make it easier to compare your polynomial with textbook series. This option is most useful when A and k are integers so the coefficients remain rational.

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