Calculator Inputs
Formula Used
Geometric series: Σ ar^n converges when |r| < 1. Sum is a / (1 - r).
p-series: Σ 1 / n^p converges when p > 1. It diverges when p ≤ 1.
Alternating p-series: Σ (-1)^n / n^p converges when p > 0. It is absolute when p > 1.
Ratio test: Let L = lim |a(n+1) / a(n)|. It converges if L < 1 and diverges if L > 1.
Root test: Let L = lim ⁿ√|a(n)|. It converges if L < 1 and diverges if L > 1.
nth-term test: If lim a(n) ≠ 0, then the series diverges.
How to Use This Calculator
- Select the series type that best matches your problem.
- Enter the coefficient, starting index, and number of terms.
- Fill the ratio, power, base, or limit fields as needed.
- Press Analyze Series.
- Read the decision, test breakdown, and graph.
- Use the CSV or PDF button to save your work.
Example Data Table
| Series | Input Type | Main Test | Expected Result |
|---|---|---|---|
| Σ 1 / n² | p-series, p = 2 | p-series test | Converges |
| Σ 1 / n | p-series, p = 1 | p-series test | Diverges |
| Σ (1/3)^n | Geometric, r = 0.3333 | Geometric test | Converges |
| Σ (-1)^n / n | Alternating p-series, p = 1 | Alternating test | Conditionally converges |
| Σ 2^n / n! | Power over factorial | Ratio test | Converges absolutely |
Understanding Series Convergence
A series adds the terms of a sequence. It may settle near a fixed value. It may also grow without bound. Convergence means the infinite sum approaches a finite number. Divergence means it does not.
Why The Test Matters
A convergence test saves time. It also shows why a result is true. A partial sum can look stable for a while. Yet the full series may still diverge. This calculator combines numeric sums with standard tests. That gives a clearer view than one method alone.
Common Tests Used
The nth term test checks the term limit. If terms do not approach zero, the series diverges. The geometric test checks a constant ratio. A geometric series converges when the absolute ratio is less than one. The p-series test checks sums like one over n to a power. It converges only when the power is greater than one.
Ratio And Root Ideas
The ratio test studies the size of the next term compared with the current term. It is strong for factorials and powers. The root test studies the nth root of the absolute term. It is useful when terms contain nth powers. Both tests are conclusive below or above one. They are inconclusive at one.
Conditional And Absolute Results
Alternating series can converge even when the positive version diverges. This is called conditional convergence. Absolute convergence is stronger. It means the series of absolute values also converges. In many applications, absolute convergence is safer because term order does not change the sum.
Using Numeric Evidence
Partial sums are helpful. They show how the series behaves over the first terms. The chart can reveal growth, oscillation, or flattening. Still, numeric evidence is not a proof by itself. Use it with a formal test. The final decision should follow the strongest valid test.
Practical Study Tips
Start with the nth term test. Then check for a known pattern. Try geometric, p-series, ratio, root, and alternating tests. Compare with a simpler benchmark when possible. Record your conclusion and reason. This helps you avoid guessing. It also makes homework steps easier to review. Clear steps build confidence before final exam practice sessions.
FAQs
1. What does convergence mean?
Convergence means the infinite sum approaches a finite value. The partial sums get closer to a stable number as more terms are added.
2. What does divergence mean?
Divergence means the infinite sum does not approach a finite value. It may grow, oscillate, or fail a required convergence condition.
3. Why is the nth-term test important?
If the terms do not approach zero, the series must diverge. This test is fast and should often be checked first.
4. When does a geometric series converge?
A geometric series converges when the absolute value of its common ratio is less than one. Otherwise, it diverges.
5. When does a p-series converge?
A p-series converges when p is greater than one. It diverges when p is equal to one or less than one.
6. What is conditional convergence?
Conditional convergence happens when an alternating series converges, but its absolute-value series diverges. The alternating harmonic series is a classic example.
7. Why can the ratio test be inconclusive?
The ratio test is inconclusive when its limit equals one. Another test, such as comparison or integral testing, is then needed.
8. Are partial sums enough for proof?
No. Partial sums give useful evidence, but a formal convergence test is needed for a reliable mathematical proof.