Power Series

What a power series is

A power series centred at a is the formal expression

f(x) = ∑_n=0^∞ aₙ (x − a)ⁿ

where each aₙ is a fixed real or complex number. Until you specify x, the series is a piece of algebra. Once you plug a value of x in, you get an infinite sum of numbers that may or may not converge. The set of x where it converges is always a disk centred at a, and inside that disk the partial sums approach a definite real number — the value of the function the series represents.

The simplest example is the geometric series:

1 + x + x² + x³ + ⋯ = 1/(1 − x) for |x| < 1.

This is the prototype: a closed-form function on the right, an infinite polynomial on the left, and an honest equality only inside a disk. Outside the disk, the right side still makes sense (1/(1 − x) at x = 2 is just −1) but the series on the left blows up.

A library of essential series

Every working analyst memorises a short list of power series and derives the rest by substitution, differentiation and integration:

• e^x = ∑ xⁿ/n!    (R = ∞)
• sin x = ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!    (R = ∞)
• cos x = ∑ (−1)ⁿ x²ⁿ/(2n)!    (R = ∞)
• 1/(1 − x) = ∑ xⁿ    (R = 1)
• −ln(1 − x) = ∑ xⁿ/n    (R = 1)
• arctan x = ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)    (R = 1)
• (1 + x)^α = ∑ (α choose n) xⁿ    (R = 1, binomial)

The arctan series falls out of the geometric one by replacing x with −x² and integrating. The logarithm series falls out of the geometric one by direct integration. This is the entire workshop of nineteenth-century analysis condensed into seven lines.

Term-by-term operations

Inside the disk of convergence, a power series behaves almost exactly like a polynomial. The four operations you can perform without checking anything beyond convergence radii:

Addition. If f(x) = ∑ aₙ xⁿ and g(x) = ∑ bₙ xⁿ have radii R_f and R_g, then f + g = ∑ (aₙ + bₙ)xⁿ with radius at least min(R_f, R_g).

Multiplication. The Cauchy product gives fg = ∑ cₙ xⁿ where cₙ = ∑_k=0ⁿ aₖ b_n−k. The radius of fg is again at least min(R_f, R_g).

Differentiation. f′(x) = ∑_n=1^∞ n aₙ xⁿ⁻¹. The radius is unchanged.

Integration. ∫f(x) dx = C + ∑_n=0^∞ aₙ xⁿ⁺¹/(n+1). The radius is unchanged.

The key theorem behind all four is uniform convergence on compact subsets of the open disk. Uniform convergence is what lets you swap limit and integral, limit and derivative, sum and product without losing equality.

Power series vs polynomials

	Polynomials	Power series
Number of terms	Finite	Infinite
Domain	All of ℝ (or ℂ)	Disk of radius R around centre
Differentiable	Yes, infinitely	Yes, infinitely (inside disk)
Closed under +, ×	Yes	Yes (inside common disk)
Closed under composition	Yes	Yes (with shrinkage of disk)
Determined by finitely many values	Yes (n+1 values for degree n)	No — but determined by all derivatives at the centre
Can represent transcendentals	No	Yes (eˣ, sin, log, arctan)

Analytic functions

A function f is analytic at a point a if there is an open interval around a on which f(x) equals a power series ∑ aₙ(x−a)ⁿ. Analyticity is much more demanding than differentiability — every analytic function is C^∞ (infinitely differentiable), but not every C^∞ function is analytic. The classic counter-example is

f(x) = e^−1/x² for x ≠ 0,   f(0) = 0.

This function is C^∞ on the whole real line, with every derivative at zero equal to zero. But its Taylor series at zero is identically 0 — yet the function is positive elsewhere. So the Taylor series fails to represent the function, and f is not analytic at 0.

What analytic functions buy you in exchange for the strictness is rigidity: an analytic function on a connected domain is determined by its values on any small arc. This is why a power series convergent in a disk extends uniquely (when it extends) to a wider domain.

The Cauchy product, illustrated

To square the geometric series:

(1 + x + x² + ⋯)² = ∑_n=0^∞ (n+1) xⁿ.

The coefficient of xⁿ in the product is the number of ways to write n = i + j with i, j ≥ 0, which is n+1. This recovers the identity 1/(1−x)² = ∑ (n+1) xⁿ, the same identity you get by differentiating the geometric series term-by-term — a useful sanity check.

Where power series appear

Generating functions in combinatorics. The number of binary trees with n internal nodes is the n-th Catalan number; its generating function C(x) = ∑ Cₙ xⁿ satisfies C = 1 + x·C², which solves to C(x) = (1 − √(1 − 4x))/(2x). Power series turn structural recurrences into algebra.

Solving differential equations. Many ODEs that resist closed-form solution yield to a power-series ansatz. Substitute y = ∑ aₙ xⁿ, expand, and equate coefficients of each power of x to get a recurrence for aₙ. This is how Frobenius and Bessel built their function families.

Defining new functions. Special functions like the Gauss hypergeometric and the error function are defined by power series. There is no simpler way to write them.

Numerical computation. Hardware sin and cos rely on truncated power series with coefficients optimised by Remez to keep the error uniform across the input range.

Common mistakes

• Adding two series outside their common disk. Term-by-term operations only hold where both series converge. ∑ xⁿ + ∑ 2ⁿ xⁿ is meaningful for |x| < 1/2, not |x| < 1.
• Substituting a series into another without checking radii. If g(0) ≠ 0, then f(g(x)) need not have the same centre as f. Re-centre carefully.
• Treating C^∞ as analytic. The bump function above is the standard reminder: smooth ≠ analytic.
• Forgetting the constant of integration. Integrating term-by-term gives a series for F(x) − F(0), not F(x) directly.
• Confusing formal manipulation with convergent equality. The series ∑ n! xⁿ can be manipulated formally as a generating function, but its disk of convergence is just the point x = 0 — there is no real-valued function it represents on any interval.