Enter Series Parameters
This calculator studies common positive decreasing series families that fit the integral test framework. It reports convergence, sample terms, partial sums, and graph-based comparisons.
Plotly Graph
The first curve shows sampled series terms. The second curve shows accumulated integral area between integer bounds.
Example Data Table
These sample inputs show how different families behave under the integral test.
| Example | Family | Parameters | Expected Outcome |
|---|---|---|---|
| 1 | Shifted p-series | c=1, p=2, k=0, n=1 | Convergent |
| 2 | Shifted p-series | c=1, p=1, k=0, n=1 | Divergent |
| 3 | Logarithmic | c=1, m=1, k=1, p=2, n=3 | Convergent |
| 4 | Exponential decay | c=3, r=0.7, k=0, n=1 | Convergent |
| 5 | Rational power | b=1, c=2, p=1.5, k=0, n=1 | Usually convergent |
Formula Used
The integral test states that a series ∑aₙ and the improper integral ∫N∞ f(x)dx behave the same way when aₙ = f(n), and f(x) is positive, continuous, and decreasing for x ≥ N.
If the improper integral converges, the series converges. If the improper integral diverges, the series also diverges. This calculator evaluates common model families with exact formulas whenever possible and numerical integration otherwise.
- Shifted p-series: ∫ c(x+k)-p dx
- Logarithmic family: ∫ c / [x(ln(mx+k))p] dx
- Exponential family: ∫ ce-r(x+k) dx
- Rational power family: numerical approximation of ∫ 1 / (b + c(x+k)p) dx
How to Use This Calculator
- Choose the series family matching your term formula.
- Enter positive parameters that keep the terms defined.
- Pick a start index where the model is continuous.
- Set a numeric upper bound for tail estimation.
- Submit the form to view the result block.
- Read the convergence decision and tail interpretation.
- Inspect the sample table and Plotly graph.
- Download the CSV or PDF report if needed.
Frequently Asked Questions
1. What does the integral test check?
It compares a positive decreasing series with a related improper integral. If both are defined from the same point onward, they share the same convergence behavior.
2. When should I trust the result?
Trust it when the function is positive, continuous, and decreasing for all x beyond your starting index. Violating those conditions can make the conclusion unreliable.
3. Why is a decreasing check included?
The integral test requires monotonic decrease after some index. The calculator samples values numerically to flag cases where your chosen model does not appear to decrease.
4. Why can a partial sum stay small while divergence occurs?
Some divergent series grow extremely slowly. Early partial sums can look harmless even though the total increases without bound over a very long range.
5. What is the difference between exact and numeric tail integrals?
Exact tails come from closed-form antiderivatives for supported families. Numeric tails use Simpson’s rule when a convenient explicit improper integral formula is not assumed.
6. Does the calculator accept every possible series?
No. It focuses on common benchmark families used in calculus courses. These models still cover many standard examples for integral test practice and interpretation.
7. Why does the logarithmic family need a larger starting index?
The logarithm must stay positive and defined. Choosing a larger start index ensures ln(mx+k) exists and keeps the denominator meaningful for the tested interval.
8. Can I use the report downloads for class notes?
Yes. The CSV captures the numerical table, while the PDF export gives a clean printable summary that works well for homework checks or revision sheets.