Calculator Inputs
The page stays single-column overall, while the form uses a responsive grid with three columns on large screens, two on medium screens, and one on mobile.
Example Data Table
These examples demonstrate how the calculator compares truncated series results against exact values.
| Function | x | a | Terms | Approximation | Exact Value | Absolute Error |
|---|---|---|---|---|---|---|
| e^x | 1.0000 | 0.0000 | 6 | 2.71666667 | 2.71828183 | 0.00161516 |
| sin(x) | 0.5000 | 0.0000 | 5 | 0.47916667 | 0.47942554 | 0.00025887 |
| cos(x) | 1.2000 | 0.3000 | 6 | 0.36302506 | 0.36235775 | 0.00066731 |
| ln(1 + x) | 0.4000 | 0.0000 | 6 | 0.33629867 | 0.33647224 | 0.00017357 |
| 1 / (1 - x) | 0.2500 | 0.0000 | 5 | 1.33203125 | 1.33333333 | 0.00130208 |
Formula Used
The calculator builds a truncated partial sum and then compares that approximation with the exact function value at the chosen input.
Core error measures
Absolute Error = |Exact Value − Approximation|
Relative Error = Absolute Error / |Exact Value|
Relative Error (%) = Relative Error × 100
Truncated partial sum
For n included terms, the approximation is the sum of the first n series terms. The next omitted term and any available remainder bound help judge how much error may remain.
Supported series
- e^x: Σ[e^a · (x − a)^n / n!]
- sin(x), cos(x): Taylor series using cyclic derivatives evaluated at a
- sinh(x), cosh(x): Taylor series using alternating derivative patterns at a
- ln(1 + x): x − x²/2 + x³/3 − x⁴/4 + …
- arctan(x): x − x³/3 + x⁵/5 − x⁷/7 + …
- 1 / (1 − x): 1 + x + x² + x³ + …
How to Use This Calculator
- Select the series function you want to approximate.
- Enter the evaluation point x where you need the estimate.
- Set the expansion center a when the selected function supports general Taylor expansion.
- Choose how many terms to include in the truncated series.
- Optionally enter a target tolerance and a scan limit to estimate the required term count.
- Press Calculate Error to display the result block above the form.
- Review the chart, term table, and remainder guidance to judge convergence quality.
- Use the CSV or PDF buttons to export the current output.
FAQs
1) What does this calculator measure?
It measures how far a truncated series approximation is from the exact function value. It reports partial sums, absolute error, relative error, and an additional estimate or bound for the remaining truncation error.
2) Why does the expansion center matter?
The center controls where the Taylor series is built. When x is near that center, the approximation often converges faster. A poor center can require many more terms for the same accuracy.
3) What is the difference between absolute and relative error?
Absolute error shows the raw gap between exact and approximate values. Relative error scales that gap by the exact value’s magnitude, making comparisons easier across large and small outputs.
4) Does the next omitted term always equal the true error?
No. It is a quick indicator, not a universal guarantee. For some alternating or geometric cases, the calculator also provides a standard remainder bound. For other cases, it is only an estimate.
5) Why are some inputs restricted?
Each power series has a convergence interval. For example, the geometric series needs |x| less than 1, while the displayed Maclaurin series for arctan(x) is intended for |x| less than or equal to 1.
6) What does the tolerance scan do?
It checks successive partial sums until the actual error falls below your target tolerance or until the scan limit is reached. This helps estimate how many terms may be needed for a desired accuracy.
7) Can I export the calculation?
Yes. The CSV export saves the summary and the full term table. The PDF export captures the visible result block and error table, which is useful for notes, reports, or classroom examples.
8) When should I increase the number of terms?
Increase terms when the absolute error remains too large, the graph shows slow convergence, or the tolerance scan does not reach your target. More terms usually improve accuracy inside the series convergence region.