Calculation history
Completed calculations will appear here.
Formula used
| Series | Term formula | Sum formula or note |
|---|---|---|
| Arithmetic | a + jd | Sₙ = n[2a + (n − 1)d]/2 |
| Geometric | arʲ | Sₙ = a(1 − rⁿ)/(1 − r) |
| Infinite geometric | arʲ | S∞ = a/(1 − r), when |r| < 1 |
| Harmonic | 1/n | Calculated by direct finite summation |
| Power | a(x + js)ᵖ | Calculated from generated terms |
| Custom | f(n, x) | Safely parsed without PHP evaluation |
The calculator generates each selected term. It then uses compensated summation for improved accuracy. Infinite geometric sums require an absolute ratio below one.
How to use this calculator
- Choose a calculation mode and series type.
- Enter the required parameters and index range.
- Select precision, rounding, and display preferences.
- Enable comparison when two series need evaluation.
- Submit the form and review every generated result.
Custom formulas may use n, j, and x. Standard functions include sqrt, sin, cos, log, and abs. Review each generated term before using important results elsewhere.
Example data
| Example | Inputs | Expected result |
|---|---|---|
| Arithmetic | a = 2, d = 3, n = 5 | 40 |
| Geometric | a = 3, r = 2, n = 4 | 45 |
| Infinite geometric | a = 8, r = 0.5 | 16 |
| Squares | n = 1 through 5 | 55 |
| Custom | 1/n², n = 1 through 4 | 1.423611… |
These examples provide quick testing values. They also demonstrate several supported calculation paths. Use sample data before entering specialised values yourself.
Frequently asked questions
What is a series sum?
A series sum adds terms from a sequence. The range may be finite or infinite. Convergence determines whether an infinite sum has one value.
How is an arithmetic series calculated?
An arithmetic series uses a constant difference. Its finite sum uses the first term and count. The calculator also supports unknown-value rearrangements for arithmetic inputs.
When does an infinite geometric series converge?
It converges when the absolute ratio is below one. Ratios outside that range do not settle. The calculator checks this condition before producing infinite results.
Can I enter my own formula?
Yes, choose the custom formula series type. Use n, j, x, constants, and supported functions. Unsafe PHP code is never evaluated by this calculator.
What is the difference between n and j?
The variable n is the actual selected index. The variable j starts at zero for generated positions. This distinction helps formulas respect custom starting indices correctly.
Why are only 1,000 terms displayed?
The limit keeps browsers responsive and tables manageable. It also prevents accidental resource-heavy calculations. Use formulas or larger server limits for extensive analysis.
Can the calculator solve a missing ratio?
Yes, finite geometric ratios use numerical solving. Some input combinations may have multiple valid roots. The entered ratio acts as the preferred starting estimate.
Why might a solved term count be decimal?
The equation can produce a mathematical non-integer count. Real sequences require a whole number of terms. Check nearby integers and compare their resulting sums carefully.
How are exports created?
CSV files contain generated table values. PDF files summarise the visible calculation and steps. Copy and print tools support faster sharing and records.