What Is the Radius of Convergence?
A power series Σcₙxⁿ is a polynomial of infinite degree, and unlike true polynomials, it doesn't have to make sense for every x. Each power series comes with a radius of convergence R: the series converges absolutely for |x| < R, diverges for |x| > R, and does something delicate exactly at |x| = R that must be checked endpoint by endpoint. Inside the radius, the series is as well-behaved as a polynomial, continuous, differentiable term by term, integrable term by term; outside, it's meaningless noise, the partial sums fly apart rather than settle.
The standard computation applies the ratio test to the series' terms, which for Σcₙxⁿ boils down to the coefficients:
R = limn→∞ |cn / cn+1| (when the limit exists)
How to Use This Calculator
Enter the coefficient formula c(n). The calculator samples |cₙ/cₙ₊₁| at increasingly large n, reads the limit, and reports R with its meaning spelled out, including the extreme verdicts R = ∞ (converges for all x, like the exponential series) and R = 0 (converges only at x = 0, the fate of Σn!xⁿ). The sampled ratios are plotted so you can watch them settle onto R. Endpoint behavior at x = ±R isn't decided by the radius, the calculator says so rather than guessing, since endpoints genuinely require separate series tests.
Worked Example
Find R for Σxⁿ/2ⁿ, i.e. cₙ = 1/2ⁿ:
|cₙ/cₙ₊₁| = (1/2ⁿ) / (1/2ⁿ⁺¹) = 2 constant, so R = 2 on the nose.
Sanity check: the series is geometric with ratio x/2, converging exactly when |x/2| < 1, i.e. |x| < 2 ✓. At the endpoints: x = 2 gives Σ1 (diverges), x = −2 gives Σ(±1) (diverges), so the interval of convergence is the open interval (−2, 2). Other series keep one or both endpoints; each is its own investigation.
The Endpoint Ritual
| Series | R | At x = R | At x = −R | Interval |
|---|---|---|---|---|
| Σxⁿ | 1 | diverges | diverges | (−1, 1) |
| Σxⁿ/n | 1 | harmonic, diverges | alternating harmonic, converges | [−1, 1) |
| Σxⁿ/n² | 1 | p = 2, converges | converges | [−1, 1] |
Three series, identical radius, three different intervals, the complete demonstration that R decides the interior and never the boundary. The endpoint checks are exactly the p-series and alternating series tests from elsewhere in this library.
Common Mistakes to Avoid
- Inverting the ratio. R = lim |cₙ/cₙ₊₁|, current over next. The reciprocal computes 1/R, silently swapping convergence and divergence regions.
- Claiming endpoint behavior from R. The radius is agnostic at |x| = R; skipping the endpoint checks loses the half-open-interval subtleties that exams love.
- Forgetting centered series. For Σcₙ(x − a)ⁿ the disk is centered at a: convergence for |x − a| < R. Same radius computation, shifted address, the disk simply moves with the expansion point.
- Mixing up coefficients and terms. The formula wants cₙ (the multiplier of xⁿ), not the whole term cₙxⁿ. For series with gaps (only even powers, say), apply the ratio test to the full terms instead.
Real-World Applications
The radius of convergence is the warranty card of every series-based computation. Physics perturbation expansions, corrections in powers of a coupling constant, are power series whose radius marks where the perturbative picture breaks down entirely; quantum electrodynamics' famous series is believed to have R = 0, meaningful only as an asymptotic expansion, a fact with deep physical interpretation. Engineering solvers that expand solutions of differential equations in power series (a standard method) can only promise their answers within R, and numerical libraries pick between series, rational, and iterative algorithms for special functions based on exactly where the argument falls relative to convergence disks.
In complex analysis, the radius acquires a beautiful geometric meaning: it's the distance from the center to the function's nearest singularity in the complex plane. That's why the innocuous real function 1/(1+x²) has R = 1 around 0 despite being smooth on the whole real line, its hidden poles at ±i, one unit away, set the limit. Series behavior on the real line is governed by geography in the complex plane: one of mathematics' great unification stories, and the reason R is computable from coefficients alone.
Frequently Asked Questions
What if the coefficient ratio has no limit?
The general formula switches to the root test: R = 1 / lim sup |cₙ|^(1/n) (Cauchy–Hadamard), which always exists. Ratio-based computation is the everyday convenience; the root test version is the theorem that never fails.
What does R = ∞ mean concretely?
Convergence for every x, the series defines an entire function. The exponential, sine, and cosine series all qualify (their factorial denominators crush every power), which is why those functions are computable at any argument by summing enough terms.
Can a series converge only at its center?
Yes: R = 0. The standard example is Σn!xⁿ, factorial growth in the coefficients overwhelms every nonzero x. Such series still matter as formal or asymptotic objects, but never as convergent sums away from the center.
Does differentiating or integrating a power series change R?
Never, term-by-term derivatives and antiderivatives keep exactly the same radius (though endpoint behavior can shift). That stability is what makes power-series manipulation so powerful: differentiate the geometric series freely inside |x| < 1 and the results remain valid there.
How does this relate to Taylor series accuracy?
A Taylor series converges to its function only within the radius, outside, adding terms makes things worse, not better. The 1/(1−x) expansion's R = 1 is why the Taylor Series Calculator's approximations for that function fall apart past |x| = 1 no matter the order: the radius is a hard wall, not a suggestion.
Why do gap series (only even powers, say) need special care?
The coefficient list alternates with zeros, so consecutive-coefficient ratios oscillate between 0 and ∞, no limit. The fix is applying the ratio test to the nonzero terms directly (treating x² as the variable) or invoking Cauchy–Hadamard with lim sup. The series for cos x is the everyday example, radius ∞ despite its gaps.
Does the radius of convergence change if I differentiate the series?
No. Term-by-term differentiation and integration of a power series preserve the radius of convergence exactly, though the behavior at the endpoints can change. This invariance is what makes power series so convenient for solving differential equations.