Calculator
Formula Used
Direct comparison convergence: If 0 ≤ an ≤ bn and Σbn converges, then Σan converges.
Direct comparison divergence: If 0 ≤ bn ≤ an and Σbn diverges, then Σan diverges.
Limit comparison: L = limn→∞ an / bn. If 0 < L < ∞, both series have the same behavior.
Reference rules: Σ1/np converges for p > 1. Geometric series converge for |r| < 1.
Log rule: Σ1/(np ln(n)q) converges for p > 1. When p = 1, it converges for q > 1.
How to Use This Calculator
- Select direct comparison, limit comparison, or both.
- Choose the model for the target term an.
- Choose a reference term bn with known behavior.
- Enter powers, shifts, coefficients, and ratios.
- Select the inequality when direct comparison is used.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to export the current calculation.
Example Data Table
| Target series | Reference series | Method | Expected conclusion |
|---|---|---|---|
| Σ 1/(n² + n) | Σ 1/n² | Limit comparison | Convergent |
| Σ 1/√n | Σ 1/n | Direct comparison | Divergent |
| Σ 3(0.4)n | Geometric rule | Reference test | Convergent |
| Σ 1/(n ln n) | Logarithmic reference | Reference test | Divergent |
Understanding the Comparison Test
The comparison test helps decide whether a positive series converges. It works by matching an unknown series with a reference series that is already understood. The method is useful when a direct sum is hard to find.
Direct Comparison
A series with terms a_n is compared with another series b_n. Both terms should be positive after some starting value. If a_n is less than or equal to b_n, and b_n converges, then a_n also converges. The smaller series cannot exceed the finite total of the larger one.
The reverse idea proves divergence. If a_n is greater than or equal to b_n, and b_n diverges, then a_n diverges too. The larger series cannot stay finite while the smaller reference grows without bound.
Limit Comparison
The limit comparison test is more flexible. It studies the limit of a_n divided by b_n. When the limit is positive and finite, both series share the same behavior. They either both converge or both diverge. This is helpful when terms are not ordered neatly.
Choosing a Reference
Pick a reference that matches the dominant growth of the term. For rational expressions, compare highest powers of n. For terms with logarithms, compare with harmonic or logarithmic series. For exponential terms, compare with a geometric series. A good reference makes the limit simple.
Using the Calculator
This calculator builds sample terms for both series. It can apply direct comparison, limit comparison, or both. It also estimates ratios at large index values. The estimate is not a proof by itself, but it supports the written reasoning. Always check positivity and the known behavior of the reference series.
Practical Notes
Use larger index values when early terms are noisy. Use tighter tolerance when ratios change slowly. If the calculator returns inconclusive, the selected comparison may not fit the series. Try another reference series with closer long-term behavior.
Checking Conditions
The test assumes nonnegative terms near infinity. A few early negative or zero terms may require separate handling. They do not control convergence alone. The tail behavior matters most. Still, the comparison must be valid for all large n. Record the inequality, the reference result, and the final conclusion. This creates a clear audit trail for every solution.
FAQs
What is the comparison test for convergence?
It is a method for testing positive series. You compare the unknown series with a known series. A smaller series than a convergent series converges. A larger series than a divergent series diverges.
What is the difference between direct and limit comparison?
Direct comparison uses an inequality between terms. Limit comparison uses the limit of a ratio. Direct comparison is stronger when a clear inequality exists. Limit comparison is easier when terms have similar long-term growth.
Can this calculator prove convergence automatically?
It gives structured evidence and applies standard rules. The final proof still depends on correct assumptions. You must verify positivity, inequalities, and exact limits when presenting a formal solution.
Which reference series should I choose?
Choose a reference with similar dominant behavior. Use p-series for powers of n. Use geometric series for exponential terms. Use logarithmic references when ln(n) controls the denominator.
What does a positive finite ratio mean?
It means a_n and b_n grow or shrink at comparable rates. In limit comparison, a positive finite ratio shows both series share the same convergence behavior.
What happens when the ratio is zero?
If a_n divided by b_n tends to zero and b_n converges, then a_n converges. The target series is eventually smaller in a strong limiting sense.
What happens when the ratio is infinite?
If a_n divided by b_n grows without bound and b_n diverges, then a_n diverges. The target series is eventually larger in a strong limiting sense.
Why is the result sometimes inconclusive?
The chosen comparison may not match the series well. The inequality may point the wrong way. The ratio may not give a decisive limit. Try a closer reference series.