Result Summary
Enter your series family and parameters. The convergence result will appear here above the form.
Calculator Inputs
Plotly Graph
The graph shows the selected positive test function f(x) from the starting index to your chosen upper x value.
Example Data Table
| Series Family | Example Series | Start n | Key Parameter | Integral Test Result | Improper Integral |
|---|---|---|---|---|---|
| a / x^p | 1 / x^2 | 1 | p = 2 | Convergent | 1 |
| a / x^p | 1 / x | 1 | p = 1 | Divergent | Infinity |
| a / (x (ln x)^p) | 1 / (x (ln x)^2) | 3 | p = 2 | Convergent | 0.910239 |
| a e^(-k x) | e^(-0.5x) | 1 | k = 0.5 | Convergent | 1.213061 |
Formula Used
1) Standard p-series
For f(x) = a / xp, the test integral is ∫n∞ a x-p dx. It converges when p > 1.
2) Shifted power form
For f(x) = a / (b x + c)p, the test integral is ∫n∞ a (b x + c)-p dx. It converges when p > 1 and b > 0.
3) Logarithmic form
For f(x) = a / (x (ln x)p), the test integral is ∫n∞ a / (x (ln x)p) dx. It converges when p > 1.
4) Exponential form
For f(x) = a e-k x, the test integral is ∫n∞ a e-k x dx. It converges when k > 0.
The calculator classifies convergence by comparing the infinite series with the matching improper integral, provided the function stays positive, continuous, and decreasing.
How to Use This Calculator
- Select the series family that best matches your problem.
- Enter the coefficient and the needed exponent or decay rate.
- Set the starting index n used for the improper integral.
- Choose a graph upper value to inspect function behavior.
- Press Evaluate Convergence to view the result above the form.
- Review the conclusion, assumptions, integral value, and graph.
- Use the export buttons to save the summary as CSV or PDF.
Frequently Asked Questions
1) What does the integral test show?
It shows whether a positive decreasing series behaves like its related improper integral. If the integral converges, the series converges. If the integral diverges, the series diverges.
2) When can I use this test safely?
Use it when the associated function is positive, continuous, and decreasing for all x at or beyond your chosen starting index. Those conditions are essential.
3) Why must the function be decreasing?
The integral test compares rectangle areas with the curve. Decreasing behavior keeps those area comparisons reliable and preserves the convergence link.
4) Does the improper integral equal the series sum?
No. The integral usually does not equal the exact series sum. It only tells you whether both expressions share the same convergence behavior.
5) Why is p > 1 important for many examples?
For p-series and related families, p > 1 makes the tail decay fast enough. When p is 1 or smaller, the tail remains too large.
6) Why must logarithmic examples start above one?
The term ln(x) must stay positive in forms like 1 / (x (ln x)^p). Starting above one avoids undefined or misleading values.
7) What does the graph add to the analysis?
The graph shows how quickly the test function falls. Faster decay usually supports convergence, while slow decay often signals divergence.
8) Can I use this for any infinite series?
No. This page handles common series families with clear integral-test rules. Arbitrary symbolic series need additional algebra or computer algebra support.