Analyze convergence radius from limits, roots, or singularities. See interval bounds around any chosen center. Download neat records, verify steps, and study examples quickly.
| Case | Center | Method | Input | Radius | Base Interval |
|---|---|---|---|---|---|
| 1 | 0 | Ratio Test | L = 2 | 1/2 | (-1/2, 1/2) |
| 2 | 3 | Root Test | ρ = 4 | 1/4 | (2.75, 3.25) |
| 3 | -1 | Nearest Singularity | d = 5 | 5 | (-6, 4) |
| 4 | 2 | Ratio Test | L = 0 | ∞ | (-∞, ∞) |
General power series: Σ an(x - c)n
Ratio test: If L = lim |an+1 / an|, then R = 1 / L.
Root test: If ρ = limsup |an|1/n, then R = 1 / ρ.
Nearest singularity: R equals the distance from c to the nearest singularity.
Interval of convergence: Start with (c - R, c + R), then test endpoints separately.
Power series appear in calculus, differential equations, and numerical analysis. A radius of convergence tells you where the series behaves well. It marks the distance from the center to the boundary of guaranteed convergence. This makes the radius a key concept for solving real problems with confidence.
A power series usually has the form sum of an(x-c)n. The value c is the center. The radius R measures how far x can move from c while the series still converges. Inside that interval, the series converges absolutely. Outside it, the series diverges. At the endpoints, you must test separately.
The ratio test is common when successive coefficients are easy to compare. If L = lim |an+1/an|, then R = 1/L when L is positive and finite. If L equals zero, the radius is infinite. If L grows without bound, the radius is zero. This method works well for factorial and exponential style coefficients.
The root test uses ρ = limsup |an|1/n. Then R = 1/ρ. This is useful when coefficients contain nth powers or products that simplify under roots. Complex analysis also gives another route. If a function has a known nearest singularity, the radius equals that distance from the center.
This calculator helps you switch between methods quickly. Enter the center, choose the method, provide the needed value, and review the computed interval. You can also mark endpoint behavior after separate testing. That makes the displayed interval more informative for study notes, homework checks, and classroom examples. Export tools and a sample table make documentation easier.
Students often confuse the radius with the interval itself. The radius is a nonnegative distance. The interval depends on both the center and endpoint tests. For example, a series centered at 3 with radius 2 gives a base interval from 1 to 5. Whether 1 or 5 belongs in the final answer depends on separate convergence checks. By keeping these pieces distinct, you avoid common mistakes. A clear workflow is simple: identify the coefficient pattern, choose ratio, root, or singularity data, compute R, write the open interval, and then test endpoints with p-series, alternating series, comparison, or integral arguments when needed.
The radius of convergence is the distance from the center of a power series to the nearest boundary where convergence can fail. It tells you how far x may move from the center while the series still converges.
No. The radius is only a distance. The interval uses that distance together with the center. Then you must test the endpoints separately to decide whether they are included.
Use the ratio test when consecutive coefficients compare neatly. It is especially helpful with factorials, exponentials, and product patterns where an+1/an simplifies well.
The root test is useful when coefficients contain nth powers or expressions that become simpler after taking the nth root. It often gives a cleaner limit than the ratio test.
If the limit is zero, the reciprocal becomes infinite. That means the radius of convergence is infinite, so the series converges for every real x.
If the limit is infinite, the reciprocal becomes zero. Then the radius of convergence is zero, and the series converges only at its center.
Yes. If the power series comes from a known function, the radius equals the distance from the center to the nearest singularity in the complex plane.
The standard radius formulas only decide behavior inside and outside the open interval. At the endpoints, convergence may differ, so separate tests are required.
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