Calculator Input
Example Data Table
| Function | a | p | Order n | Known Coefficient |
|---|---|---|---|---|
| e^x | 1 | — | 0 | 1 |
| e^x | 1 | — | 4 | 1/24 |
| sin(x) | 1 | — | 5 | 1/120 |
| cos(x) | 1 | — | 6 | -1/720 |
| ln(1+x) | 1 | — | 4 | -1/4 |
| (1+x)^3 | 1 | 3 | 2 | 3 |
Formula Used
The Maclaurin series of a function is f(x) = Σ anxn, where an = f(n)(0) / n!. This calculator applies closed-form coefficient rules for standard analytic functions.
- e^(a·x): an = a^n / n!
- sin(a·x): nonzero only for odd powers
- cos(a·x): nonzero only for even powers
- sinh(a·x): odd-power coefficients equal a^n / n!
- cosh(a·x): even-power coefficients equal a^n / n!
- 1 / (1 - a·x): an = a^n
- ln(1 + a·x): an = (-1)^(n+1) a^n / n for n ≥ 1
- arctan(a·x): odd-power coefficient is (-1)^((n-1)/2) a^n / n
- (1 + a·x)^p: an = C(p,n) a^n with generalized binomial coefficients
How to Use This Calculator
- Choose a supported function model from the dropdown.
- Enter the scaling parameter a.
- Enter p only for the binomial model.
- Select the coefficient order n you need.
- Choose how many series terms should be generated.
- Enter an x value to inspect term contributions.
- Press Calculate Coefficient to show results above the form.
- Use the export buttons to save the generated table.
Frequently Asked Questions
1) What does a Maclaurin coefficient represent?
A Maclaurin coefficient is the multiplier of x^n in the expansion around x = 0. It tells how strongly that power contributes to the local behavior of the function.
2) Why are some coefficients zero?
Functions with symmetry often lose either odd or even powers. For example, cosine keeps only even powers, while sine keeps only odd powers.
3) What is the difference between coefficient and term value?
The coefficient is a_n only. The term value is a_n x^n after substituting your chosen x value. Both help analyze approximation quality.
4) Why does the calculator show a partial sum?
The partial sum estimates the function using the first few terms. Comparing it with the actual value shows how accurate your truncated series is.
5) When is the result unavailable in the real domain?
Some choices break domain rules. Examples include ln(1 + a·x) with nonpositive input, or fractional powers of negative bases in real arithmetic.
6) What does the convergence note mean?
It explains where the series behaves reliably. Some series converge everywhere, while others require |a·x| less than 1 for stable expansion results.
7) Can I use negative or decimal parameters?
Yes. The calculator accepts negative and decimal values for a, x, and p. That makes it useful for many scaled and generalized expansions.
8) What is the graph showing?
The Plotly graph displays generated coefficients and evaluated term values by order. It helps you see decay, alternation, and dominant terms quickly.