Complex Sequence Calculator

Calculator Form

Choose a sequence model, set the displayed range, and generate terms. Expressions support n, prev_re, prev_im, and functions such as sin, cos, sqrt, log, exp, min, and max.

Generation Method

Start Index

Displayed Terms

Decimal Precision

Explicit Real Part u(n)

Example: cos(n), 1/n, sqrt(n), exp(-0.2*n)

Explicit Imaginary Part v(n)

Example: sin(n), n/5, log(n), (-1)^n

Initial Real Part at n = 1

Used only for the recursive method.

Initial Imaginary Part at n = 1

Set the starting complex value a₁.

Recursive Real Formula

Use n, prev_re, and prev_im.

Recursive Imaginary Formula

Example creates a rotating, shrinking sequence.

Geometric First Term Real Part

This is the real part of a₁.

Geometric First Term Imaginary Part

This is the imaginary part of a₁.

Ratio Real Part

Real component of the common ratio r.

Ratio Imaginary Part

Imaginary component of the common ratio r.

Example Data Table

Example shown for the geometric model with a₁ = 1 + 0i and r = 0.7 + 0.4i.

n	Example a_n	\|a_n\|	arg(a_n)°
1	1.0000 + 0.0000i	1.0000	0.0000
2	0.7000 + 0.4000i	0.8062	29.7449
3	0.3300 + 0.5600i	0.6500	59.4898
4	0.0070 + 0.5240i	0.5240	89.2347

Formula Used

Explicit model: a_n = u(n) + i·v(n), where u and v are user-defined real-valued formulas.
Recursive model: a₁ = α + iβ, then Re(a_n) = f(n, prev_re, prev_im) and Im(a_n) = g(n, prev_re, prev_im).
Geometric model: a_n = a₁ · r^n-1, where a₁ and r are complex numbers.
Magnitude: |a_n| = √(Re(a_n)² + Im(a_n)²).
Phase angle: arg(a_n) = atan2(Im(a_n), Re(a_n)).
Displayed partial sum: S_k = Σ a_n across the generated range.
Geometric sum identity: when r ≠ 1, S_N = a₁(1 - r^N) / (1 - r).

How to Use This Calculator

Select a generation method: explicit, recursive, or geometric.
Enter the start index, number of displayed terms, and decimal precision.
Provide the real and imaginary formulas or the complex parameters for the chosen model.
Click Calculate Complex Sequence to show the results below the header and above the form.
Review the summary cards, inspect the Plotly graph, and read the full term table.
Use the CSV or PDF buttons to save the displayed results for reporting or teaching.

Frequently Asked Questions

1) What does this calculator evaluate?

It generates complex sequence terms and reports each term’s real part, imaginary part, magnitude, phase angle, running displayed sum, and consecutive-term ratio.

2) Which sequence types are supported?

You can work with explicit formulas, first-order recursive rules, and complex geometric sequences. That combination covers many classroom and self-study examples.

3) Which variables may I use inside formulas?

Explicit formulas use n. Recursive formulas can use n, prev_re, and prev_im. Standard functions such as sin, cos, sqrt, log, exp, min, and max are also supported.

4) What does the phase angle mean?

The phase angle is the direction of each complex term in the Argand plane. It is measured in degrees using the atan2 function.

5) Why is the ratio column sometimes blank?

The first displayed term has no previous term inside the displayed range. A ratio also becomes undefined when the previous complex term equals zero.

6) Is the convergence hint a formal proof?

No. It is a practical interpretation of the displayed terms only. Use it as a visual clue, not as a rigorous mathematical proof.

7) Can I use this for teaching and assignment checking?

Yes. The summary cards, graph modes, and exports make it useful for demonstrations, worked examples, revision, and quick validation of manually computed terms.

8) Does the calculator export only the visible results?

Yes. The CSV and PDF exports include the currently displayed rows and summary values, so the saved file matches the screen output.