Numerical Series and Convergence Tests

Infinite Sum

The sum 1 + 1/2 + 1/4 + 1/8 + ... — can an infinite sum be finite? Intuition says "no," but mathematics shows: yes, if the terms decrease fast enough.

A numerical series is the formal notation Σₙ₌₁^∞ aₙ. Its partial sum is Sₙ = a₁ + a₂ + ... + aₙ. The series converges if the sequence {Sₙ} has a finite limit S. Then S = Σₙ₌₁^∞ aₙ.

Necessary condition for convergence: If the series converges, then aₙ → 0. The converse is not true: the series Σ 1/n (the harmonic series) diverges, although 1/n → 0.

Geometric Series

Σₙ₌₀^∞ qⁿ = 1/(1-q) for |q| < 1. Diverges for |q| ≥ 1.

Proof: Sₙ = (1-qⁿ⁺¹)/(1-q) → 1/(1-q) for |q| < 1.

This is a fundamental series: annuity rates, business valuation problems (the series of discounted cash flows) are expressed through it.

Convergence Tests for Series with Nonnegative Terms

Comparison test: If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges.

D'Alembert's test: lim(n→∞) aₙ₊₁/aₙ = L. Converges for L < 1, diverges for L > 1.

Cauchy's (root) test: lim(n→∞) ⁿ√aₙ = L. Converges for L < 1, diverges for L > 1.

Raabe's test: lim n(aₙ/aₙ₊₁ - 1) = L. Converges for L > 1, diverges for L < 1. Applied when D'Alembert's test gives L = 1.

Maclaurin–Cauchy integral test: if f(x) is nonnegative and monotonically decreasing, then Σf(n) and ∫f(x)dx behave the same way.

Absolute and Conditional Convergence

The series Σaₙ converges absolutely if Σ|aₙ| converges. Absolute convergence → convergence.

Leibniz's test (for alternating series): If |aₙ| monotonically decreases to zero, then Σ(-1)ⁿaₙ converges.

The series Σ(-1)ⁿ/n converges conditionally (to ln 2), but not absolutely.

Riemann's Theorem

A conditionally convergent series can be rearranged so as to obtain any given sum (or +∞, or -∞). This is a surprising result: the order of terms is crucial for conditionally convergent series.

Rate of Decrease and Rate of Convergence

Not only the fact of convergence, but also the rate of decrease of the terms is essential for computational practice. If aₙ ~ C/nᵅ, the series converges for α > 1. If aₙ decreases exponentially (aₙ ≤ Mrⁿ, r < 1) — the series converges rapidly. D'Alembert's test lim aₙ₊₁/aₙ = L < 1 means that the terms decrease "almost geometrically" with ratio L.

Series in Computations

Practice: computing π through the Leibniz series π/4 = 1 − 1/3 + 1/5 − ... converges too slowly (≈10⁶ terms for 6 digits). Much more efficient is Machin's formula: π/4 = 4arctan(1/5) − arctan(1/239), where the arctan expansions converge rapidly. Modern algorithms for calculating π are based on Ramanujan’s quadratically convergent formulas and the Borwein algorithm — the use of series with properly chosen rate of convergence.

Abel's Theorem on Boundary Behavior

On the boundary of the circle of convergence |x − a| = R, the behavior of the series is not determined in general. Abel's theorem: if the power series Σcₙxⁿ converges at x = R (the boundary point), then its sum is continuous at x = R (from the left). This allows one to compute the value of functions at the boundary of convergence via the limit from the inside.

Corollary: ln 2 = 1 − 1/2 + 1/3 − 1/4 + ... This series converges (Leibniz's test), and by Abel's theorem, its sum equals the value of ln(1+x) at x = 1.

In probability theory, the moment generating function E[eˢˣ] = Σ μₙsⁿ/n! is a power series whose coefficients are the moments of the distribution. The method of moments for recovering a distribution from statistical data is based on this expansion.

Arithmetic of Convergent Series

If the series Σaₙ and Σbₙ converge to A and B, then Σ(aₙ + bₙ) = A + B and Σ(caₙ) = cA (linearity). The product of series (Cauchy product) Σcₙ where cₙ = Σₖ₌₀ⁿ aₖbₙ₋ₖ converges to AB if at least one of the original series converges absolutely. Without the condition of absolute convergence, the product may diverge — for example, the product of the series Σ(-1)ⁿ/√(n+1) with itself diverges. This limitation is fundamental for applications in probability theory, where convolution of distributions (summation of independent random variables) requires absolute convergence of the corresponding series.

Question for thought: The series Σ1/n² = π²/6 — "Basel problem," solved by Euler in 1735. Why does the sum of a quadratic series contain π, although π is a "geometric" number?

Comparison Tests and the Asymptotic Test

For numerical series with positive terms: if aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges; if aₙ ≥ cₙ and Σcₙ diverges, then Σaₙ diverges. Limit comparison test: if aₙ/bₙ → L ∈ (0, ∞), then the series behave the same way. Example: 1/(n² + 3n) ~ 1/n² as n → ∞, so the first series converges together with the second. Maclaurin–Cauchy integral test: Σf(n) converges if and only if ∫₁^∞ f(x) dx converges (for decreasing f). This allows one to reduce convergence of a series to convergence of an improper integral. P-series: Σ1/nᵖ converges for p > 1 and diverges for p ≤ 1.