Calculator Inputs
Formula Used
A Maclaurin series is a Taylor series centered at zero. Its coefficients come from derivatives evaluated at zero. Each coefficient is divided by the matching factorial value.
How to Use This Calculator
Select a built-in function or enter a supported expression. Choose the term range, output style, and evaluation value. Press calculate to generate coefficients, errors, graphs, and tables.
- Choose a function and enter its parameters.
- Select terms, degree limits, parity, and precision.
- Enter an x-value and desired error tolerance.
- Review the polynomial, table, graph, and convergence status.
- Copy or export the completed calculation when needed.
Common Maclaurin Series
These standard expansions provide useful checks for calculated results. Their convergence regions differ for logarithmic and rational functions. Always review endpoints before using boundary values in practice.
| Function | First terms | Convergence |
|---|---|---|
| ex | 1 + x + x²/2! + x³/3! + … | All real x |
| sin x | x - x³/3! + x⁵/5! - … | All real x |
| cos x | 1 - x²/2! + x⁴/4! - … | All real x |
| ln(1+x) | x - x²/2 + x³/3 - … | -1 < x ≤ 1 |
| arctan x | x - x³/3 + x⁵/5 - … | -1 ≤ x ≤ 1 |
| (1+x)p | 1 + px + p(p-1)x²/2! + … | |x| < 1 generally |
Example Data
Use these presets to test different calculator capabilities. Each example highlights terms, approximation, or convergence behavior. Change values afterward to explore related functions and scenarios.
| Example | Function | Settings | Purpose |
|---|---|---|---|
| 1 | ex | 6 terms, x = 0.5 | General approximation |
| 2 | sin x | 4 nonzero terms, x = 1 | Alternating error |
| 3 | ln(1+x) | 8 terms, x = 0.5 | Finite radius |
| 4 | (1+x)1/2 | 6 terms, x = 0.25 | Binomial expansion |
| 5 | exp(2x)cos(3x) | Degree 10 | Composite function |
Frequently Asked Questions
1. What is a Maclaurin series?
It is a Taylor series centered specifically at zero. Derivatives at zero determine every polynomial coefficient. The expansion can approximate functions near zero very effectively.
2. How many terms should I use?
More terms usually improve accuracy inside the convergence interval. The required count depends on x and tolerance. Use the tolerance result to select an efficient degree.
3. Why are some coefficients zero?
Symmetry often removes either odd or even powers. Sine uses odd powers while cosine uses even powers. Zero coefficients remain mathematically valid but may be hidden.
4. What is the difference between Taylor and Maclaurin series?
Taylor series may use any chosen expansion center. Maclaurin series always use zero as their center. Their formulas otherwise follow the same derivative-based construction method.
5. What does radius of convergence mean?
It measures how far the series converges from zero. Values inside this radius generally produce valid approximations. Endpoint convergence must often be checked using separate tests.
6. What is the next-term error estimate?
It uses the first omitted term as an error guide. This is especially useful for suitable alternating series. The term magnitudes should decrease under the required conditions.
7. What is the Lagrange remainder?
It bounds truncation error using a derivative maximum. Enter a valid bound M for the interval. The calculator then applies the standard remainder inequality directly.
8. Can I enter a custom function?
Yes, supported patterns and ordinary polynomials are accepted. Arbitrary code is rejected to protect the server. Use the displayed examples when formatting custom mathematical expressions.
9. Why might the approximation fail?
The selected point may lie outside convergence limits. The original function might also be undefined there. Review warnings, intervals, and endpoint notes before using results.