Maclaurin Series Calculator

Estimate function values with clean power series expansion. Inspect terms, errors, and convergence details. Compare curves with downloadable outputs and helpful examples today.

Calculator Inputs

Function

Evaluation Point x

Highest Order n

Graph Start x

Graph End x

Graph Step

Decimal Precision

Supports error analysis Shows partial sums Plots exact and approximate curves Includes CSV and PDF export

Example Data Table

Function	x	Order n	Approximation Insight
e^x	1.0	6	Gives a close estimate to e.
sin(x)	0.5	7	Odd powers dominate this alternating approximation.
cos(x)	1.2	8	Even powers improve stability near zero.
ln(1+x)	0.3	10	Works well inside the convergence interval.
arctan(x)	0.7	9	Alternating odd terms refine the angle estimate.

Formula Used

The Maclaurin series is the Taylor series centered at zero:

f(x) = Σ [ f⁽ⁿ⁾(0) / n! ] xⁿ, for n = 0 to ∞

This calculator builds a truncated polynomial:

P_N(x) = Σ [ f⁽ⁿ⁾(0) / n! ] xⁿ, for n = 0 to N

Error metrics are computed as:

Absolute Error = | Exact Value − Approximation |
Relative Error % = (Absolute Error / |Exact Value|) × 100
Partial Sum = cumulative total after each included term

How to Use This Calculator

Select the target function from the list.
Enter the x value where you want the estimate.
Choose the highest order n for the polynomial.
Set graph start, graph end, and graph step values.
Pick the decimal precision for displayed results.
Press the calculate button to show results above the form.
Review the approximation, exact value, errors, and convergence note.
Use the term table, graph, CSV, or PDF export as needed.

Frequently Asked Questions

1. What does a Maclaurin series calculator do?

It approximates a function near zero using polynomial terms from the function’s derivatives at zero. It also compares the approximation against the exact value when available.

2. Why do more terms usually improve accuracy?

Additional terms capture more curvature and behavior of the original function. For many functions, partial sums get closer to the exact value as order increases within the convergence region.

3. Is the series always valid for every x value?

No. Some Maclaurin series converge only inside a limited interval. For example, geometric and logarithmic forms often require |x| less than 1 for reliable convergence.

4. What is the difference between exact value and approximation?

The exact value comes from the original function formula. The approximation comes from the truncated Maclaurin polynomial, which uses only a finite number of terms.

5. Why are some terms shown as zero?

Many functions use only even or odd powers in their Maclaurin expansion. Terms with missing powers naturally become zero and still help explain the full order structure.

6. What does the graph compare?

The graph plots the exact function and the Maclaurin approximation over your chosen x range. This helps you see where the truncated series tracks closely and where it drifts.

7. When should I use a smaller x range?

Use a smaller range when the function converges slowly or has a limited convergence interval. Near the expansion center, polynomial approximations usually behave much better.

8. What can I do with CSV and PDF export?

CSV export is useful for spreadsheets and deeper analysis. PDF export creates a printable report of the visible page, including the summary and term table.