1
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2.9994
1.9998
2.9994
1
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2.9994
1.9998
2.9994
The Radius of Convergence Calculator estimates the radius of convergence $$R$$ of a power series from its first few coefficients using the ratio test. Given a power series:
$$\sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + \cdots$$
the radius of convergence determines the interval around the center $$c$$ where the series converges absolutely.
The ratio test provides the standard method for computing the radius of convergence. It states that:
$$R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|$$
when this limit exists. The series converges absolutely for $$|x - c| < R$$ and diverges for $$|x - c| > R$$. At the boundary $$|x - c| = R$$, convergence must be tested separately.
In practice, we often know only a few coefficients. By computing the ratios $$|a_0/a_1|$$, $$|a_1/a_2|$$, and $$|a_2/a_3|$$, we obtain successive approximations to the limit. If the ratios are converging, the last ratio $$|a_2/a_3|$$ typically gives the best estimate. This calculator shows all three ratio estimates plus their average.
The Cauchy–Hadamard theorem provides an alternative formula using the root test:
$$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}$$
This is more general than the ratio test (it always yields a result), but the ratio test is easier to compute by hand and usually sufficient for series arising in practice.
When $$R = \infty$$, the series converges for all $$x$$ (entire functions like $$e^x$$, $$\sin(x)$$). When $$R = 0$$, the series converges only at $$x = c$$. Most power series encountered in applications have a finite, positive radius of convergence.
The radius of convergence determines the domain of validity for power series solutions of differential equations, Laurent series in complex analysis, generating functions in combinatorics, and transfer functions in control theory. It also connects to the analytic structure of functions—$$R$$ equals the distance from the center to the nearest singularity in the complex plane.
Enter the first four coefficients $$a_0, a_1, a_2, a_3$$ of your power series. The calculator computes the ratio $$|a_n / a_{n+1}|$$ for each consecutive pair, giving three estimates of $$R$$. The best estimate uses the last available ratio, which is closest to the limiting value.
If the ratio estimates are converging to a common value, that value is a reliable estimate of $$R$$. If they are diverging or erratic, more coefficients would be needed. A very large value (99999) indicates a zero coefficient, suggesting possible infinite convergence radius. The series converges for $$|x - c| < R$$ and diverges for $$|x - c| > R$$.
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The ratios 1, 2, ~3 are growing toward infinity, consistent with R = ∞ for eˣ. With more coefficients, the ratios would continue to grow (|aₙ/aₙ₊₁| = n+1 → ∞).
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All ratios equal 1, confirming R = 1 exactly. The series 1 + x + x² + x³ + ⋯ converges for |x| < 1.
The radius of convergence tells you the range of $$x$$-values for which a power series representation is valid. Using the series outside this range produces meaningless, divergent results. It is essential for determining the domain of series solutions in differential equations and for understanding the analytic behavior of functions.
If $$a_{n+1} = 0$$, the ratio $$|a_n / a_{n+1}|$$ is undefined (shown as 99999). This can happen when the series has missing terms (e.g., $$\sin(x)$$ has no even-powered terms). In such cases, skip the problematic ratio and rely on the other estimates, or use the root test instead.
For many common series, 4 coefficients give a reasonable estimate. The accuracy depends on how quickly the ratios $$|a_n/a_{n+1}|$$ converge. For the geometric series (all coefficients equal), even 2 coefficients give the exact answer. For slowly converging ratios, more terms would improve the estimate.
The ratio test is inconclusive at the boundary. The series may converge, diverge, or converge conditionally at $$x = c + R$$ and $$x = c - R$$. Each endpoint must be tested individually using other convergence tests such as the alternating series test or comparison test.
In the complex plane, $$R$$ equals the distance from the center $$c$$ to the nearest singularity of $$f(z)$$. For example, $$1/(1-x)$$ has a singularity at $$x = 1$$, so $$R = 1$$ when centered at $$c = 0$$. This beautiful connection links analysis to the geometry of complex functions.
The ratio test uses $$R = \lim |a_n/a_{n+1}|$$, while the root test uses $$1/R = \limsup |a_n|^{1/n}$$. The root test is more general (it always gives an answer via limsup), but the ratio test is easier to compute. When both limits exist, they give the same value of $$R$$.
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