This sample uses the exponential series around c = 0, where ez ≈ 1 + z + z²/2! + z³/3! + ...
| n | Coefficient an | Meaning | Sample Entry |
|---|---|---|---|
| 0 | 1 | Constant term | 1 |
| 1 | 1 | Linear term | 1 |
| 2 | 1/2 | Quadratic term | 0.5 |
| 3 | 1/6 | Cubic term | 0.1666666667 |
| 4 | 1/24 | Fourth-order term | 0.0416666667 |
| 5 | 1/120 | Fifth-order term | 0.0083333333 |
Power Series Formula
S(z) = Σ an(z - c)n, for n = 0, 1, 2, ...
The calculator evaluates a finite partial sum from the coefficients you provide.
Nth Partial Sum
SN(z) = Σn=0N an(z - c)n
Term Magnitude
|an(z - c)n| helps assess how quickly contributions shrink or grow.
Estimated Radius Of Convergence
Root estimate: R ≈ 1 / max(|an|1/n) for entered n ≥ 1
Ratio estimate: R ≈ median(|an / an+1|) for consecutive nonzero coefficients
Estimated Truncation Error
A practical error hint is taken from the last term magnitude. If recent term decay looks geometric, a tail estimate is used.
- Enter the center c using its real and imaginary parts.
- Enter the evaluation point z where you want the series value.
- Paste the coefficients in order from a0 upward.
- Choose the maximum number of terms and your preferred precision.
- Select a radius estimate method and stopping mode.
- Press the calculate button to show the result above the form.
- Review the summary cards, full term table, and Plotly graph.
- Use the CSV or PDF buttons to export the computed output.
1) What does this calculator evaluate?
It evaluates a finite complex power series partial sum at a chosen complex point. It also estimates radius behavior, error size, and convergence clues from your entered coefficients.
2) How should I enter coefficients?
Enter one coefficient per line or separate them with commas. Use ascending order: a0, a1, a2, and so on. Formats like 2, -3.5, i, and 2+4i are accepted.
3) What is the center c?
The center is the expansion point of the series. Each power is built from (z - c), so changing c changes every computed term and the final partial sum.
4) Why is the radius only estimated?
A true radius of convergence usually needs the full infinite coefficient pattern. This tool only sees the finite list you provide, so it returns a useful numerical estimate.
5) What does the truncation error mean?
It is a practical hint for the leftover tail after the last included term. Smaller values usually mean the displayed partial sum is more stable for your chosen setup.
6) Can I model Taylor or Maclaurin series here?
Yes. A Maclaurin series uses c = 0. A Taylor series around another point uses that point as c. Enter the corresponding coefficients in order.
7) Why do partial sums sometimes oscillate or spiral?
Complex terms can rotate in the plane while changing magnitude. That combination can make partial sums move irregularly before stabilizing, especially near the convergence boundary.
8) What do the CSV and PDF exports include?
They include the summary metrics and the computed term table shown after submission. This makes it easier to save numerical work, compare runs, and share results.