Infinite, Numerical and Harmonic Series

An infinite (numerical) series is an expression a₁ + a₂ + a₃ + … with infinitely many terms, and it converges if the sequence of its partial sums approaches a finite limit. In brief: the series 1 + 1/2 + 1/4 + 1/8 + … converges to 2, while the harmonic series 1 + 1/2 + 1/3 + 1/4 + … diverges to infinity, even though its terms tend to 0. A necessary condition for convergence is that a_n → 0, but it is not sufficient.

Adding infinitely many numbers can give a finite number, and this fact has surprised mathematicians since ancient times. The tool for making sense of such sums is the sequence of partial sums. This guide defines convergence, explains the necessary condition and the divergence test, studies the harmonic series and the p-series, covers geometric and telescoping series and alternating series, with a table of standard series and their behaviour.

What is an infinite series and when does it converge?

The series Σ a_n (the sum from n = 1 to ∞) has the partial sums S_N = a₁ + … + a_N. The series converges to S if S_N → S as N → ∞, and then we write Σ a_n = S. If the partial sums do not approach a finite number, the series diverges.

Example 1: convergence

Σ 1/2ⁿ = 1/2 + 1/4 + 1/8 + … has the partial sums 1 − 1/2^N, which approach 1. So the sum is 1. This is a geometric series with a₁ = 1/2 and r = 1/2.

Example 2: telescoping

Σ 1/(n(n + 1)) = Σ (1/n − 1/(n + 1)). The partial sum is S_N = 1 − 1/(N + 1), which tends to 1. So the sum is 1.

Example 3: divergence

Σ n = 1 + 2 + 3 + … has partial sums N(N + 1)/2, which grow without bound. The series diverges.

What is the necessary condition for convergence?

If Σ a_n converges, then a_n → 0. The reason is that a_N = S_N − S_N−1 → S − S = 0. Therefore, if a_n does not tend to 0, the series diverges (the divergence test). For example, Σ (n/(n + 1)) diverges since the terms tend to 1, and Σ (−1)ⁿ diverges, since the terms do not tend to 0.

The converse is false: terms tending to 0 do not guarantee convergence. The harmonic series is the standard counterexample.

What is the harmonic series?

The harmonic series is 1 + 1/2 + 1/3 + 1/4 + …, and it diverges. A classical argument, due to Nicole Oresme in the fourteenth century, groups the terms: 1 + 1/2 + (1/3 + 1/4) + (1/5 + … + 1/8) + … Each group is at least 1/2: 1/3 + 1/4 > 1/4 + 1/4 = 1/2, and 1/5 + 1/6 + 1/7 + 1/8 > 4 · 1/8 = 1/2. With infinitely many groups, the sum exceeds any bound.

The divergence is extremely slow: the partial sum H_N is approximately ln N + 0.5772 (the Euler–Mascheroni constant γ ≈ 0.5772). To get H_N > 10, one needs N ≈ 12,367 terms; to exceed 20, about 2.7 × 10⁸.

What are p-series?

The series Σ 1/n^p converges if p > 1 and diverges if p ≤ 1. For p = 1 it is the harmonic series. For p = 2: 1 + 1/4 + 1/9 + 1/16 + … = π²/6 ≈ 1.6449, a result found by Euler in 1735 (the Basel problem). For p = 3: the sum is about 1.2021 (Apéry’s constant). For p = 1/2: 1 + 1/√2 + 1/√3 + … diverges.

Proof sketch for p > 1 by comparison with an integral: Σ_n≥2 1/n^p ≤ ∫₁^∞ x^−p dx = 1/(p − 1). So for p = 2, the sum is at most 1 + 1 = 2, and it is finite.

What are alternating series?

An alternating series has terms of alternating signs: Σ (−1)ⁿ⁺¹b_n with b_n > 0. The Leibniz test says: if b_n decreases and tends to 0, the series converges. The error of a partial sum is smaller than the first omitted term.

Example 4

The alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + … converges (by the Leibniz test), and its sum is ln 2 ≈ 0.6931. The partial sums: 1, 0.5, 0.833, 0.583, 0.783, 0.617, … oscillate around the limit. After 100 terms, the error is below 1/101 ≈ 0.0099.

Example 5

1 − 1/3 + 1/5 − 1/7 + … = π/4 (the Leibniz–Gregory series). It converges very slowly: 1000 terms give an error of about 0.0005.

Absolute and conditional convergence

A series converges absolutely if Σ |a_n| converges. Then Σ a_n converges too. The alternating harmonic series converges, but not absolutely (since Σ 1/n diverges), so it converges conditionally. For conditionally convergent series, rearranging the terms can change the sum, which does not happen for absolutely convergent series (Riemann’s rearrangement theorem).

Table of standard series

Where do infinite series come from?

Archimedes summed a geometric series to find the area of a parabolic segment. Zeno’s paradoxes raised the question of whether infinitely many positive quantities can have a finite sum. Oresme proved the divergence of the harmonic series in the fourteenth century. Euler, Bernoulli and others calculated sums of series in the seventeenth and eighteenth centuries, often without rigorous criteria, and Cauchy and Weierstrass set the theory on a firm basis in the nineteenth century.

Where are they used today?

Series are used to define and compute functions such as e^x, sin x and ln x (power series), and to solve differential equations. Calculators and computers evaluate these functions with truncated series. In probability, expected values of discrete distributions are series. The study of the harmonic series appears in the analysis of algorithms (for instance, the average case of quicksort has the harmonic number H_n).

Which mistakes are common?

Concluding convergence from a_n → 0

The terms 1/n tend to 0, but the series diverges.

Adding infinitely many terms as if they were finite

Rearranging or grouping terms of a divergent or conditionally convergent series can change or create a sum. For example, grouping 1 − 1 + 1 − 1 + … as (1 − 1) + (1 − 1) + … gives 0, but grouping as 1 + (−1 + 1) + … gives 1.

Mixing up the sequence and the series

1/n is a sequence with the limit 0. Σ 1/n is a series with no finite sum.

Using the geometric formula for |r| ≥ 1

The formula a₁/(1 − r) holds only for |r| < 1.