Calculator Input
Plotly Graph
The chart compares individual terms with cumulative partial sums for the first displayed terms.
Example Data Table
| Series Type | Input Example | Partial Sum Example | Convergence Result |
|---|---|---|---|
| Geometric | a = 3, r = 0.5, n = 8 | 5.9765625 | Convergent, because |r| < 1 |
| P-Series | a = 1, p = 2, n = 10 | 1.5497677312 | Convergent, because p > 1 |
| Alternating P-Series | a = 1, p = 1, n = 10 | 0.6456349206 | Conditionally convergent |
| Telescoping | a = 4, n = 10 | 3.6363636364 | Convergent by cancellation |
Formula Used
Term: a · rn-1
Finite sum: Sn = a(1 - rn) / (1 - r), when r ≠ 1
Infinite sum: S = a / (1 - r), when |r| < 1
Term: a / np
Converges only when p > 1
Term: (-1)n-1 · a / np
Converges when p > 0
Absolutely convergent when p > 1
Term: a · (1/n - 1/(n+1))
Finite sum: Sn = a(1 - 1/(n+1))
Infinite sum: S = a
How the Calculator Determines Convergence or Divergence
This calculator uses direct rules for standard series families. Geometric series use the ratio condition |r| < 1. P-series use the rule p > 1. Alternating p-series use the alternating series test and absolute convergence check. Telescoping series rely on term cancellation.
It also checks whether the terms approach zero. If the terms fail that basic requirement, the infinite series must diverge.
How to Use This Calculator
- Choose the series type that matches your expression.
- Select whether you want a finite partial sum or an infinite-series check.
- Enter the coefficient, ratio, exponent, and term count as needed.
- Press Calculate Series to display the result above the form.
- Review the convergence decision, infinite-sum estimate, chart, and term table.
- Use the CSV or PDF buttons to export your result summary.
FAQs
1) What is a convergent series?
A convergent series is an infinite sum whose partial sums approach one fixed number. If the running totals settle toward a limit, the series converges.
2) How does this calculator determine the convergence or divergence of the series?
It checks the selected family and applies its standard rule. Geometric series use |r| < 1, p-series use p > 1, alternating p-series use p > 0, and telescoping series use cancellation.
3) What is the difference between a partial sum and an infinite sum?
A partial sum adds only the first n terms. An infinite sum is the limit of those partial sums as n grows without bound, provided that limit exists.
4) When does a geometric series converge?
A geometric series converges only when the absolute value of the common ratio is less than one. If |r| is one or larger, the infinite series diverges.
5) When does a p-series converge?
A p-series of the form 1/np converges only when p is greater than one. It diverges for p equal to one or any smaller value.
6) What does conditionally convergent mean?
It means the original alternating series converges, but the series formed from absolute values diverges. This commonly happens for alternating p-series with 0 < p ≤ 1.
7) Why is the infinite sum sometimes shown as an estimate?
Some series, especially p-series and alternating p-series, do not have a simple closed form here. The calculator uses a high-term numerical approximation and reports a practical estimate.
8) Can I use this calculator to check homework or lecture examples?
Yes. It is useful for checking partial sums, convergence decisions, and visual trends. Still, always write your own steps and theorem names when submitting academic work.