Calculator
How to use this calculator
- Select the convergence method that matches your series form.
- Enter the needed parameters, such as p, q, r, or a limit value.
- Choose how many terms you want in the sample plot and table.
- Press Submit to place the result below the header and above the form.
- Read the verdict, criterion, notes, and sample table together.
- Use the CSV or PDF buttons if you want an exportable record.
Formula used
- Geometric series: Σ a·r^(n−1) converges when |r| < 1, and then S = a / (1 − r).
- p-series: Σ 1 / n^p converges exactly when p > 1.
- Alternating p-series: Σ (−1)^(n−1) / n^p converges absolutely for p > 1 and conditionally for 0 < p ≤ 1.
- Logarithmic power series: Σ 1 / (n^p (ln n)^q) converges when p > 1, or when p = 1 and q > 1.
- Ratio test: L = lim |a_(n+1) / a_n| gives convergence for L < 1, divergence for L > 1, and no decision for L = 1.
- Root test: L = lim |a_n|^(1/n) gives convergence for L < 1, divergence for L > 1, and no decision for L = 1.
- Nth-term test: if lim a_n ≠ 0, the series diverges. If lim a_n = 0, another test is still needed.
Example data table
| Series form | Main test | Parameter choice | Verdict |
|---|---|---|---|
| Σ 3(0.5)^(n−1) | Geometric | r = 0.5 | Convergent |
| Σ 1 / n^2 | p-series | p = 2 | Convergent |
| Σ 1 / n | p-series | p = 1 | Divergent |
| Σ (−1)^(n−1) / n | Alternating | p = 1 | Conditionally convergent |
| Σ 1 / (n (ln n)^2) | Integral/comparison | p = 1, q = 2 | Convergent |
| lim |a_(n+1)/a_n| = 1.3 | Ratio | L = 1.3 | Divergent |
FAQs
1. What does convergence mean for a series?
A series converges when its partial sums approach one finite number. If the partial sums fail to settle to a finite value, the series diverges.
2. Why is the nth-term test often inconclusive?
Because a zero term limit is only necessary. Many divergent series, such as the harmonic series, still have terms that approach zero.
3. When should I use the ratio test?
Use it for factorials, exponentials, and many power-like expressions. It is especially strong when consecutive terms simplify neatly after division.
4. When is the root test better than the ratio test?
The root test is often cleaner when terms contain n-th powers, such as c^n or expressions already raised to 1/n patterns.
5. Can an alternating series converge without absolute convergence?
Yes. That is called conditional convergence. The alternating harmonic series is the classic example.
6. Why does the graph show partial sums?
Partial sums visually reveal whether values appear to stabilize, oscillate within a narrowing band, or drift away. They support intuition, not proof.
7. What happens at p = 1 for a p-series?
The series becomes harmonic and diverges. That boundary value is the exact cutoff between convergence and divergence for Σ 1 / n^p.
8. Can this calculator replace a full written proof?
No. It helps with fast classification and study support, but coursework and exams may still require a justified method and complete reasoning.