Radius of Convergence

The Taylor series of 1/(1 + x²) about a converges only within a distance R of a. Move the degree, the point x₀ and the centre, and watch the real graph and the complex plane together: the disc on the plane decides where the curve can be trusted.

Worked example — Radius of Convergence

Radius of convergence, R	1.000000
Where the series converges	−1.000000 < x < 1.000000
How far out x₀ is, in radii	1.1000
Degree-80 polynomial at x₀	1,121.975
The function at x₀	0.4524887
Error of the polynomial	1,121.522
Most the error can be	1,515.693
Average of the partial sums up to degree 80	1.774285

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Radius of Convergence

The radius of convergence of a power series Σ c_n(x − a)ⁿ is the number R for which the series converges when |x − a| < R and diverges when |x − a| > R. For 1/(1 + x²), R is the distance from a to the nearest point of the complex plane where the function blows up, i or −i, so R = √(1 + a²).

What each symbol means

R: the radius of convergence about a, |a − i| = √(1 + a²)
q: (a − x)/(a − i); its size |x − a|/R says how many radii x lies from the centre

Where the formula comes from

Where the coefficients come from. Split 1/(1 + z²) into (1/2i)[1/(z − i) − 1/(z + i)]. Each piece is a geometric series about a: 1/(z − i) = 1/((a − i) + (z − a)) = Σ (−1)ⁿ(z − a)ⁿ/(a − i)ⁿ⁺¹, which converges exactly when |z − a| < |a − i|. The other piece is the same with a + i, at the same distance, so both stop at R = √(1 + a²).
Why the error is a power of q. A geometric series stopped after N + 1 terms leaves q^N+1 times the whole as its remainder, so f(x) − T_N(x) = Im[q^N+1/(x − i)] exactly. Its size is at most |q|^N+1/√(1 + x²): it dies like a geometric sequence when |q| < 1, grows like one when |q| > 1, and on |q| = 1 it neither dies nor grows.

No trouble anywhere on the real line, and the series still dies at 1 🖖

Press Just past 1. 1/(1 + x²) is as tame as a function gets: positive, never above 1, with a derivative of every order at every real number. Yet degree 80 about 0 gives 1,121.975 at x = 1.1, where the curve is 0.4524887, and adding terms makes it worse: every two degrees multiply the error by 1.21. Press Just inside 1: 0.9 is the same distance from 1 on the other side, and the same degree is off by 9.777×10⁻⁵. Nothing at x = 1 explains the difference; the curve passes through it at ½ without a kink. The lower panel does. Let x be a complex number z, and 1/(1 + z²) is infinite where z² = −1, at i and −i, both exactly 1 from the centre.

The radius is the distance to i, wherever you centre the series 🖖

Drag the centre a along the real axis in the lower panel. The disc grows and shrinks so that its edge always passes through i and −i, R = √(1 + a²), and the shaded band on the real graph above is the disc’s shadow on the real axis. Press Centred at 1: R = 1.414214, so the series converges from −0.414214 to 2.414214, and at x = 2, where the series about 0 gave 9.671×10²³, degree 80 is off by 2.728×10⁻¹³. Neither end of that interval is special on the real line. Each is where a circle round 1 through the two poles happens to cross the real axis.

On the circle itself the series neither settles nor runs away 🖖

Press On the edge. At x = 1 the series about 0 becomes 1 − 1 + 1 − 1 + …, Grandi’s series: degree by degree its partial sums are 1, 1, 0, 0, 1, 1, …, and they never settle and never grow. The function there is ½. The last row of the results averages the partial sums, and the average, 0.5061728 at degree 80, closes on exactly that ½.

Problem solved in full

The series fails at x = 1. Does its integral? 4 steps
Integrate the degree-80 polynomial from 0 to 1, term by term, and compare it with the true area under 1/(1 + x²). At x = 1 the series itself never settles. What happens to the area?

Open this problem in the tool
1. The true area is an arctangent:
  
  $\int_{0}^{1} \frac{d x}{1 + x ^{2}} = arctan 1 = \frac{π}{4} = 0.7853982$
2. Term by term, each x^2k becomes 1/(2k + 1):
  
  $\int_{0}^{1} T_{80} d x = \int_{0}^{1} (1 - x^{2} + x^{4} - \dots + x^{80}) d x = 1 - \frac{1}{3} + \frac{1}{5} - \dots + \frac{1}{81} = 0.7914948$
3. The difference is the integral of the error, which the geometric series gives exactly, and 1/(1 + x²) lies between ½ and 1 on [0, 1]:
  
  $\int_{0}^{1} (T_{80} - f) d x = \int_{0}^{1} \frac{x ^{82}}{1 + x ^{2}} d x \in (\frac{1}{166}, \frac{1}{83}) = (0.0060241, 0.0120482); 0.0060967$
4. That bound goes to 0 as the degree grows, so the alternating series of odd reciprocals converges, and to π/4:
  
  $0 < \int_{0}^{1} \frac{x ^{2 k + 2}}{1 + x ^{2}} d x < \frac{1}{2 k + 3} \to 0 \Rightarrow 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{π}{4}$
Answer

The area converges where the series did not: 0.7914948 against π/4 = 0.7853982, and every extra pair of degrees closes the gap. It converges slowly, the error after n terms being close to 1/(4n), so an error of a millionth takes about 250,000 terms. It also converges only conditionally, since 1 + ⅓ + ⅕ + … diverges, which is exactly the kind of series Riemann showed can be re-ordered to add up to anything. In the same 1854 essay he noted that series in powers of a variable converge absolutely, single values of the variable excepted; x = 1 is one of those values.

$0.7914948 - 0.7853982 = 0.0060967 \approx \frac{1}{4 \times 41}; 1 0^{- 6} needs \approx 250, 000 terms$

Learning path

Infinite series and convergence

References (2)

The theorem the page draws, that a function holomorphic on a region can be expanded about any point into a power series whose radius is at least the distance to the region's boundary, and this page's own example, 1/(z² + 1) about 0, radius 1 because of the singularities at ±i: M. Beck, G. Marchesi, D. Pixton and L. Sabalka, A First Course in Complex Analysis, edition 1.6 (2025), open textbook, Corollary 8.1.10 and Example 8.1.11.
Riemann's remark, in the essay where he gave the rearrangement theorem, that series in rising powers of a variable belong to the class that converges absolutely, single values of the variable excepted: B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe", Habilitationsschrift, Göttingen 1854; published by R. Dedekind in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), §3. Transcribed by D. R. Wilkins, Trinity College Dublin.

By the 42Math team · Reviewed against published sources. Corrections welcome.