Calculator Inputs
Formula Used
Let u = b(x - c). The calculator converts the selected family into a standard series in u, then multiplies each coefficient by A.
- A / (1 - u): Σun, for n ≥ 0 and |u| < 1.
- A / (1 + u): Σ(-1)nun, for n ≥ 0 and |u| < 1.
- Aeu: Σun / n!, for n ≥ 0.
- A sin(u): Σ(-1)nu2n+1 / (2n+1)!, for n ≥ 0.
- A cos(u): Σ(-1)nu2n / (2n)!, for n ≥ 0.
- A ln(1 + u): Σ(-1)n+1un / n, for n ≥ 1.
- A ln(1 - u): -Σun / n, for n ≥ 1.
- A arctan(u): Σ(-1)nu2n+1 / (2n+1), for n ≥ 0.
Each truncated polynomial is the partial sum formed from the first chosen nonzero terms.
How to Use This Calculator
- Choose a supported function family.
- Enter A, b, and the expansion center c.
- Select how many nonzero terms to display.
- Enter a value of x for approximation testing.
- Set graph limits and submit the form.
- Review the series, coefficient table, and sample errors.
- Use the CSV or PDF buttons to save results.
Example Data Table
| Example Function | Series Around c = 0 | Radius Note |
|---|---|---|
| 1 / (1 - x) | 1 + x + x2 + x3 + ... | |x| < 1 |
| ex | 1 + x + x2/2! + x3/3! + ... | All real x |
| sin(x) | x - x3/3! + x5/5! - ... | All real x |
| ln(1 + x) | x - x2/2 + x3/3 - ... | |x| < 1 |
| arctan(x) | x - x3/3 + x5/5 - ... | |x| < 1 |
FAQs
1. What does this calculator find?
It builds a truncated power series for a supported function family. It also shows coefficients, convergence notes, a comparison graph, and exportable tables for study or teaching.
2. Is this a fully symbolic series engine?
No. It handles common standard families with scaling and shifting. That makes the output clear, fast, and useful for most calculus practice problems.
3. Why is the approximation sometimes inaccurate?
A truncated series only keeps a limited number of terms. Accuracy drops when x moves farther from the center or near the edge of convergence.
4. What does the center c control?
The center c determines where the polynomial is built. The approximation is usually strongest near that center because higher-order powers stay smaller there.
5. Why do some exact values show as undefined?
Some functions are not defined for every input. Logarithms need positive arguments, and rational functions fail when their denominators become zero.
6. What does the convergence note mean?
It tells you where the infinite series is mathematically reliable. For geometric and logarithmic families, that interval matters strongly for good approximations.
7. What is included in the CSV file?
The coefficient export stores term number, exponent, coefficient, and readable term text. The sample export stores x, exact value, approximation, and difference.
8. Can I use this for classroom demonstrations?
Yes. The layout is simple, the tables are readable, and the plot shows how the partial sum tracks the original function across a chosen interval.