A Direct Comparison Test Calculator helps determine whether an infinite series converges or diverges by comparing it with another known series. This method is commonly used in calculus and mathematical analysis to study the behavior of sequences and series.
The tool simplifies the process by automatically checking whether a given series meets the comparison test conditions, making it valuable for students and professionals working with infinite series in mathematics, engineering, and physics.
Formula of Direct Comparison Test Calculator
The Direct Comparison Test is based on the following principle:
If 0 ≤ aₙ ≤ bₙ for all n ≥ N (some positive integer N), then:
- If the series Σbₙ converges, then Σaₙ also converges.
- If the series Σaₙ diverges, then Σbₙ also diverges.
Where:
- aₙ = terms of the first series
- bₙ = terms of the second (comparison) series
- Σaₙ = sum of the first series
- Σbₙ = sum of the second series
- N = starting index for the comparison
This test works by comparing a given series with another series whose convergence or divergence is already known.
Commonly Used Series for Comparison
This table provides a quick reference for common series that are frequently used in the Direct Comparison Test.
| Series Type | General Form | Convergence/Divergence |
|---|---|---|
| p-Series (p > 1) | Σ 1/n^p | Converges |
| p-Series (p ≤ 1) | Σ 1/n^p | Diverges |
| Geometric Series ( | r | < 1) |
| Geometric Series ( | r | ≥ 1) |
| Harmonic Series | Σ 1/n | Diverges |
| Alternating Series | Σ (-1)^n aₙ (if decreasing) | Converges |
This reference helps users find suitable comparison series for their calculations.
Example of Direct Comparison Test Calculator
Determine whether the series Σ (1 / (n² + 1)) converges using the Direct Comparison Test.
Step 1: Identify the series terms
The given series has terms:
aₙ = 1 / (n² + 1)
Step 2: Choose a comparison series
A known convergent p-series is:
bₙ = 1 / n², which converges because p = 2 > 1.
Since:
1 / (n² + 1) ≤ 1 / n² for all n ≥ 1,
we apply the Direct Comparison Test.
Step 3: Apply the test
Since Σbₙ = Σ (1 / n²) converges, and 0 ≤ aₙ ≤ bₙ, the given series Σaₙ also converges.
Thus, Σ (1 / (n² + 1)) is a convergent series.
Most Common FAQs
Use this test when you can compare a given series with a known convergent or divergent series and prove the inequality 0 ≤ aₙ ≤ bₙ.
If a direct comparison does not work, try the Limit Comparison Test, which uses limits to compare series instead of direct inequalities.
Yes, if you compare a given series with a known divergent series and show that aₙ ≥ bₙ for all n ≥ N, then the given series also diverges.