Calculator Inputs
Example Data Table
| Function | n | x | f⁽ⁿ⁾(0) | Coefficient | Nth Term |
|---|---|---|---|---|---|
| e^x | 4 | 0.5 | 1 | 1 / 24 | 0.0026041667 |
| sin(x) | 5 | 0.5 | 1 | 1 / 120 | 0.0002604167 |
| cos(x) | 4 | 0.5 | 1 | 1 / 24 | 0.0026041667 |
| 1 / (1 - x) | 3 | 0.25 | 6 | 1 | 0.015625 |
Formula Used
The Maclaurin series is a Taylor series centered at zero:
f(x) = Σ [f⁽ⁿ⁾(0) / n!] xⁿ
The nth term is:
Tₙ(x) = [f⁽ⁿ⁾(0) / n!] xⁿ
The coefficient is:
aₙ = f⁽ⁿ⁾(0) / n!
An optional Lagrange style remainder estimate is:
|Rₙ(x)| ≤ M |x|ⁿ⁺¹ / (n + 1)!
How to Use This Calculator
Select the function type first. Enter the order n. Add the value of x where the term should be evaluated. Use the custom derivative field when the function is not listed. Enter a maximum order to build a partial sum table. Add a derivative bound and point value when a remainder estimate is needed. Press calculate. The result appears above the form and below the page header.
Maclaurin Series Nth Term Guide
What This Calculator Does
A Maclaurin series rewrites a function as a power series centered at zero. It helps you study local behavior without graphing every point. The nth term shows the single term used at a chosen order. This calculator focuses on that term, the matching coefficient, and the partial sum.
Why Derivatives Matter
The general term is built from derivatives at zero. For a function f, the nth Maclaurin term is f^(n)(0) x^n divided by n factorial. When that value is known, the coefficient becomes f^(n)(0) divided by n factorial. The calculator accepts either the derivative value or the ready coefficient. This keeps the tool useful for classroom work and quick checking.
Supported Function Patterns
Many common functions follow familiar patterns. The exponential function uses coefficients equal to one over n factorial. Sine uses only odd powers. Cosine uses only even powers. The geometric form uses repeated powers when the function fits one over one minus x. The custom mode lets you enter your own derivative value when your function is not listed.
Partial Sums and Remainders
The partial sum estimates the function near zero. More terms often improve accuracy inside the interval of convergence. A remainder estimate can also be entered. This is useful when a problem gives a bound for the next derivative. The tool reports an optional Lagrange style bound when enough values are supplied.
Steps for Accurate Use
Use the calculator by choosing a function type first. Then enter the order n and the evaluation value x. Add a derivative value for custom work. Add a maximum order when you want a partial sum. Press calculate to see the result above the form. You can also export the result as CSV or PDF for notes.
Learning Value
This calculator is best for checking algebra, building tables, and learning term patterns. It does not replace proof work. Always confirm the interval of convergence for a full series answer. Also remember that factorials grow quickly, so large orders may create very small terms. The worked table shows how terms change as n increases. That pattern often reveals whether a series uses all powers or only selected powers. Try several x values near zero. Compare each partial sum carefully. Small changes can show convergence speed, sign changes, and numerical stability in daily practice.
FAQs
What is a Maclaurin series?
A Maclaurin series is a Taylor series centered at zero. It represents a function as powers of x using derivative values taken at zero.
What is the nth term?
The nth term is the single series term for a selected order. It equals f⁽ⁿ⁾(0) divided by n factorial, multiplied by x raised to n.
Can I use this for custom functions?
Yes. Choose the custom option and enter the derivative value at zero for your selected order. The calculator then builds the term from that value.
Why is the factorial important?
The factorial scales each derivative value. It usually makes higher order terms smaller, especially when x is close to zero.
Does this show the full series?
It shows the selected nth term and a partial sum table. A full series still needs a general pattern and convergence discussion.
What is the coefficient?
The coefficient is f⁽ⁿ⁾(0) divided by n factorial. It is the multiplier placed before x raised to n in the series.
What is a remainder estimate?
A remainder estimate gives a possible error bound after stopping at a chosen order. This calculator uses an optional Lagrange style bound.
Why does sine skip even powers?
The even derivatives of sine at zero become zero. So their terms vanish, leaving only odd powers in the Maclaurin series.