Power series

A power series is an infinite series of form sum_{n=0}^{infty} a_n (x - x_0)^n with coefficients a_n and center x_0. Power series represent analytic functions and are the central tool of real and complex analysis. Every holomorphic function locally coincides with its Taylor series.

Radius of convergence

The radius of convergence R of a power series is the (possibly infinite) number such that the series:

• converges absolutely for |x - x_0| < R;
• diverges for |x - x_0| > R;
• behavior for |x - x_0| = R requires specific analysis.

Cauchy-Hadamard formula

Radius of convergence is computed as:

1/R = limsup_{n -> infty} |a_n|^(1/n)

Equivalently, if the limit exists, R = lim |a_n / a_{n+1}| (ratio test). For Maclaurin series of elementary functions, R is easy: R(geometric) = 1, R(e^x) = infty, R(ln(1+x)) = 1, R(sin/cos) = infty.

Classical examples

• Geometric series: sum x^n = 1/(1-x), R = 1.
• Exponential: e^x = sum x^n/n!, R = infty.
• Sine: sin(x) = sum_{k=0} (-1)^k x^(2k+1)/(2k+1)!, R = infty.
• Cosine: cos(x) = sum_{k=0} (-1)^k x^(2k)/(2k)!, R = infty.
• Logarithm: ln(1+x) = sum_{n=1} (-1)^(n+1) x^n/n, R = 1.
• Binomial: (1+x)^a = sum C(a,n) x^n, R = 1 (for non-integer a).
• Arctan: arctan(x) = sum_{n=0} (-1)^n x^(2n+1)/(2n+1), R = 1.

Absolute and uniform convergence

Inside convergence disk |x - x_0| < R, series converges ABSOLUTELY. On any compact contained in this disk, convergence is UNIFORM. So: can integrate and differentiate term by term, swap limit and sum, etc. Sum is holomorphic function.

Boundary behavior

For |x - x_0| = R, series may: (

• diverge; (
• converge but not absolutely (e.g., ln(1+x) at x=1: 1 - 1/2 + 1/3 - ... = ln 2); (
• converge absolutely. Abel theorem: if converges at boundary point, continuous from inside up to that point. Examples: arctan(

1) = pi/4 (Leibniz), ln(2) (Mercator).

Power series operations

• Sum: (sum a_n x^n) + (sum b_n x^n) = sum (a_n + b_n) x^n.
• Cauchy product: (sum a_n x^n)(sum b_n x^n) = sum c_n x^n with c_n = sum_{k=0}^{n} a_k b_{n-k}.
• Termwise differentiation: f'(x) = sum n a_n x^(n-1).
• Integration: integral f = sum a_n x^(n+1)/(n+1).
• Composition: f(g(x)) substitutes g series (if g(0) = 0).

Taylor series

For f infinitely differentiable at x_0, Taylor series is f(x) = sum f^(n)(x_0)/n! (x - x_0)^n. If f is analytic at x_0, series converges to f in a neighborhood. NOT all infinitely differentiable functions are analytic: example f(x) = e^(-1/x^2) for x != 0, f(0) = 0 has Taylor series at 0 identically zero, but doesn't equal f.

Truncation error

To estimate error of partial sum S_N = sum_{n=0}^{N} a_n x^n:

• Lagrange: error <= max|f^(N+1)(c)| / (N+1)! * |x - x_0|^(N+1).
• For alternating series with |a_n+1| <= |a_n|, error <= |a_{N+1} x^{N+1}| (Leibniz criterion).
• For geometric series with ratio q < 1: error = |q^{N+1}| / (1 - q).

Applications

• Numerical computation of elementary functions (sin, cos, exp, log).
• ODE solutions: power series as Frobenius method.
• Combinatorics: generating functions are formal series.
• Physics: perturbative expansions (small parameter).
• Complex analysis: representation of holomorphic functions.
• Number theory: generating functions (Riemann zeta).
• Probability: characteristic functions, moment generators.

History

Madhava (XIV century) develops series for arctan and sine. Newton (1665-1672) introduces general binomial series. Brook Taylor 1715 formalizes Taylor theorem. Cauchy 1821 (Cours d Analyse) rigorously defines convergence. Hadamard 1888 proves convergence radius formula. Karl Weierstrass 1880 uses series as principal analysis tool.

Analytic functions and continuation

A function is said to be analytic at a point if, in a neighbourhood of that point, it coincides with the sum of a convergent power series. This property is surprisingly strong: analytic functions are infinitely differentiable and completely determined by their values in a small neighbourhood. Elementary functions such as the exponential, sine, cosine and logarithm are all analytic in their domains, which is why they can be represented exactly by power series.

A deep concept is analytic continuation. The geometric series 1/(1-x) converges only for |x| < 1, but the function 1/(1-x) is defined and smooth everywhere except at x = 1. The series is only a local "window" onto a wider function, which can be extended beyond the radius of convergence by following the function itself. In the complex field this continuation is unique, a principle underlying fundamental results such as the definition of the Riemann zeta function on the whole complex plane.

Power series in practical computation

Power series are not just a theoretical tool: they are how calculators and computers evaluate transcendental functions. When you ask for the sine or exponential of a number, the processor does not consult a table but sums a finite number of terms of a power series, stopping when the truncation error falls below the required precision. A preliminary argument-reduction step brings the value into a small interval where the series converges rapidly.

To maximize efficiency, techniques such as Chebyshev economization are used, which reorganize the coefficients to reduce the number of terms needed. Power series also appear as generating functions in combinatorics, where the coefficients encode entire sequences of numbers, and in solving differential equations by series. From graphics to scientific computing, every evaluation of a non-elementary function goes, behind the scenes, through a truncated power series.

Updated July 2026