Calculator Inputs
Choose a series model, enter values, then expand and evaluate instantly.
Plotly Convergence Graph
The chart compares successive partial sums with the exact value. It currently shows a sample exponential series until you submit your own values.
Example Data Table
| Series | Expression | x | Terms | Approximate | Exact |
|---|---|---|---|---|---|
| Binomial Expansion | (2 + 0.5x)^5 | 1.2 | 6 | 118.81376 | 118.81376 |
| Geometric Series | 3 / (1 - 0.2x) | 1.5 | 7 | 4.284777 | 4.285714 |
| Exponential Maclaurin | 2·e^(0.8x) | 1 | 6 | 4.450261 | 4.451082 |
| Sine Maclaurin | 1·sin(1.2x) | 0.9 | 5 | 0.881958 | 0.881958 |
| Cosine Maclaurin | 1.5·cos(0.7x) | 1.1 | 5 | 1.076866 | 1.076866 |
Formula Used
Binomial Expansion
(a + bx)n = Σ C(n,k) an-k(bx)k, for k = 0 to n.
Geometric Series
a / (1 - bx) = Σ a(bx)k, for k = 0 to ∞, valid when |bx| < 1.
Exponential Maclaurin
a·ebx = Σ a(bk/k!)xk, for k = 0 to ∞.
Sine Maclaurin
a·sin(bx) = Σ (-1)ka(b2k+1/(2k+1)!)x2k+1.
Cosine Maclaurin
a·cos(bx) = Σ (-1)ka(b2k/(2k)!)x2k.
How to Use This Calculator
- Select the series family you want to expand.
- Enter coefficient a and multiplier or ratio b.
- Choose the x value where the series should be evaluated.
- Set the number of terms for the truncated expansion.
- Enter exponent n when using binomial mode.
- Pick decimal places for cleaner output formatting.
- Click Expand and Evaluate to view results above the form.
- Review the graph, table, and export buttons for reporting.
FAQs
1. What can this calculator expand?
It expands binomial, geometric, exponential, sine, and cosine series. Each option builds a truncated series, evaluates it at your chosen x value, and compares it with an exact value when available.
2. What is the difference between approximate and exact values?
The approximate value comes from the selected number of terms. The exact value comes from the original closed form. Their difference shows how much truncation error remains.
3. Why do I see a geometric convergence warning?
An infinite geometric series converges only when |bx| is less than 1. When that condition fails, partial sums still calculate, but they do not approach the infinite-series value.
4. Does binomial mode support any exponent?
This version is designed for nonnegative integer exponents. That keeps the expansion finite, stable, and easy to compare with the exact polynomial value.
5. What does the Plotly graph show?
The graph tracks how the partial sum changes as each new term is added. A second line marks the exact value, making convergence speed easy to inspect visually.
6. Can I use negative coefficients or negative x values?
Yes. Negative coefficients and negative x values are supported. Just remember that geometric convergence still depends on the magnitude of bx, not its sign alone.
7. Why does adding more terms usually improve accuracy?
Each additional term captures more of the original function’s behavior. For convergent series, partial sums move closer to the exact value as the number of included terms grows.
8. What do the CSV and PDF exports include?
Exports include the active result summary, selected inputs, formula note, and the example table. They are useful for coursework, documentation, and quick sharing.