Calculator Inputs
Formula Used
f(x) ≈ Σ from k = 0 to n of [f(k)(a) / k!] · (x - a)k
This calculator evaluates the chosen function and its derivatives at the center a, converts each derivative into a Taylor coefficient, and then sums all terms from k = 0 through the selected order n.
The approximation error is computed as |exact value − partial sum|. Percent error is computed as: absolute error / |exact value| × 100, whenever the exact value is nonzero.
The next omitted term estimate is the (n + 1)th Taylor term evaluated at x. It is useful as a quick size check for the first missing contribution beyond the chosen order.
How to Use This Calculator
- Select a supported function or choose the custom polynomial option.
- Enter the expansion center a and the evaluation point x.
- Choose the truncation order n and the number of decimal places.
- If using a polynomial, enter comma-separated coefficients in ascending powers.
- Press Expand Series to generate the approximation, error values, and detailed term table.
- Use the export buttons to save the computed summary and term-by-term results as CSV or PDF.
Example Data Table
| Function | Center a | x | Order n | Approximation | Exact value | Absolute error |
|---|---|---|---|---|---|---|
| e^x | 0 | 1 | 5 | 2.71666667 | 2.71828183 | 0.00161516 |
| sin(x) | 0 | 0.5 | 5 | 0.47942708 | 0.47942554 | 0.00000154 |
| ln(1 + x) | 0 | 0.3 | 6 | 0.26233190 | 0.26236426 | 0.00003236 |
FAQs
1) What does this calculator actually compute?
It builds a finite Taylor polynomial around the chosen center, evaluates that polynomial at x, and compares the result with the exact function value.
2) What is the difference between Taylor and Maclaurin series?
A Maclaurin series is simply a Taylor series centered at a = 0. Any other center value produces a standard Taylor expansion.
3) Why does changing the center matter?
The center determines where derivatives are measured and where the approximation is strongest. A point closer to x often improves accuracy with fewer terms.
4) What does the convergence radius mean here?
It estimates how far from the center the infinite series is normally valid before hitting a nearby singularity. Inside that radius, approximations are usually more reliable.
5) Why can the percent error show N/A?
If the exact value is zero or extremely close to zero, percent error becomes unstable or undefined. The absolute error remains the safer measure in that case.
6) How do I enter a custom polynomial?
Provide coefficients in ascending powers, such as 3, -2, 0, 5 for 3 − 2x + 5x³. The calculator then derives and evaluates the polynomial exactly.
7) Does a higher order always guarantee better results?
Usually near the center, yes. Far from the center, improvement can slow down, fail, or behave poorly when the point lies near or beyond the convergence boundary.
8) What do the CSV and PDF downloads include?
They include the summary metrics and the full term table, so you can save, review, print, or share the approximation details outside the calculator.