Calculator Inputs
Use ratio, root, or coefficient estimation for the series Σ an(x - c)n.
Formula Used
For a power series of the form Σ an(x - c)n, the radius of convergence is:
- Ratio test: If L = lim |an+1 / an|, then R = 1 / L.
- Root test: If α = limsup |an|1/n, then R = 1 / α.
- Inside the radius: If |x - c| < R, the series converges absolutely.
- Outside the radius: If |x - c| > R, the series diverges.
- At endpoints: If |x - c| = R, test each endpoint separately.
Determine the Convergence or Divergence of the Series
This calculator helps determine convergence or divergence by comparing the test point distance |x - c| against the radius R.
- If |x - c| < R, the series converges absolutely.
- If |x - c| > R, the series diverges.
- If |x - c| = R, the result depends on the endpoint series, so separate testing is required.
Use the endpoint assumption menus when you already know whether the left or right endpoint converges.
How to Use This Calculator
- Select the calculation method that matches your series information.
- Enter the center c from the power series expression.
- Provide the ratio limit, root limsup, or a coefficient list.
- Add a test value of x to check convergence or divergence.
- Choose endpoint assumptions if you already know endpoint behavior.
- Press the calculate button to view the radius, interval, and graph.
- Download the result summary as CSV or PDF if needed.
Example Data Table
| Series | Method | Center c | Input | Radius R | Interval |
|---|---|---|---|---|---|
| Σ n((x - 2)n / 3n) | Ratio test | 2 | L = 1/3 | 3 | (-1, 5) |
| Σ ((x + 1)n / n) | Root test | -1 | α = 1 | 1 | (-2, 0), endpoints separate |
| Σ (x - 4)n / n! | Ratio test | 4 | L = 0 | ∞ | (-∞, ∞) |
| Σ n!(x)n | Ratio test | 0 | L = ∞ | 0 | {x = 0} |
FAQs
1. What does the radius of convergence tell me?
It tells you how far from the center the power series remains convergent. Inside that radius, the series converges absolutely. Outside it, the series diverges.
2. How does this determine convergence or divergence?
It compares the distance |x - c| to the radius R. Smaller means convergence, larger means divergence, and equality means the endpoint needs separate testing.
3. When should I use the ratio test option?
Use it when you can compute or already know the limit of |a(n+1)/a(n)|. It works especially well for factorials, exponentials, and many rational coefficient patterns.
4. When is the root test better?
Use the root test when coefficient powers are easier to simplify than ratios. It is also helpful when limsup behavior is clearer than consecutive-term behavior.
5. Can this calculator handle infinite radius?
Yes. If the ratio limit or root limsup is zero, the radius becomes infinite, meaning the series converges for every real x.
6. What if the radius is zero?
Then the series converges only at its center c. Any test point different from c lies outside the convergence region and gives divergence.
7. Why are endpoints treated separately?
At |x - c| = R, absolute convergence is not guaranteed. Many series behave differently at the two endpoints, so each endpoint must be tested on its own.
8. Is the coefficient-list method exact?
No. It estimates the radius from the tail of the supplied coefficients. It is useful for exploration, but symbolic limits remain more reliable when available.