Definition of Series
A series is the sum of the terms of a sequence. While a sequence is simply an ordered list of numbers — such as 1, 1/2, 1/4, 1/8, … — a series asks a deeper question: what happens when you add all those numbers together? Writing 1 + 1/2 + 1/4 + 1/8 + … turns the sequence into a series. The idea sounds straightforward, but it leads to one of the most profound concepts in mathematics: can infinitely many numbers add up to a finite value? To make sense of adding infinitely many terms, mathematicians introduce the concept of partial sums. Instead of trying to add every term at once, you add the first term, then the first two, then the first three, and so on, watching whether the running total settles toward a fixed number. If that running total approaches a specific value — called the limit — the series is said to converge and its sum is well-defined. If the running total grows without bound or oscillates forever, the series diverges and no finite sum exists. Understanding series is essential across mathematics, physics, and engineering. Taylor and Maclaurin series let us approximate complicated functions like sin(x) and e^x using simple polynomials. Fourier series decompose sound waves and electrical signals into sums of sinusoids. Even calculating the decimal expansion of numbers like 1/3 = 0.333… is secretly an infinite series. The formal study of series bridges the intuitive idea of repeated addition with the rigorous machinery of limits.
- Start with a Sequence
- Form the Partial Sums
- Define the Infinite Series
- Convergence vs. Divergence
- Sigma Notation
Start with a Sequence
Every series begins with a sequence of terms. A sequence is a function whose domain is the positive integers: a₁, a₂, a₃, …, aₙ, … For example, the sequence aₙ = 1/2ⁿ gives the terms 1/2, 1/4, 1/8, 1/16, …
Form the Partial Sums
The nth partial sum Sₙ is defined as the sum of the first n terms of the sequence: Sₙ = a₁ + a₂ + a₃ + … + aₙ. For the sequence above: S₁ = 1/2, S₂ = 3/4, S₃ = 7/8, S₄ = 15/16. Notice that each partial sum is getting closer and closer to 1.
Define the Infinite Series
The infinite series is written using sigma notation as ∑_{n=1}^{∞} aₙ. Its value — if it exists — is defined as the limit of the partial sums: ∑_{n=1}^{∞} aₙ = lim_{n→∞} Sₙ. This limit is not computed by 'adding infinitely many numbers all at once'; it is the limit of the sequence S₁, S₂, S₃, … If this limit exists and is finite, the series converges.
Convergence vs. Divergence
A series converges if lim_{n→∞} Sₙ = L for some finite real number L. Otherwise it diverges. The harmonic series 1 + 1/2 + 1/3 + 1/4 + … is a famous example that diverges: its partial sums grow without bound, even though the individual terms shrink to zero. This shows that terms going to zero is necessary but not sufficient for convergence.
Sigma Notation
The compact way to write an infinite series is ∑_{n=1}^{∞} aₙ. The symbol ∑ (capital Greek sigma) means 'sum,' the expression below it gives the starting index, and ∞ above it indicates the sum continues forever. The index variable n is a 'dummy variable' — it could equally well be called k or i. The expression aₙ is the general term, the rule telling you what to add.
The Key Equation
- \sum_{n=1}^{\infty} a_n — Infinite series (dimensionless (or same units as aₙ)): The series formed by summing all terms of the sequence starting at n = 1.
- a_n — General term (dimensionless): The nth term of the underlying sequence; the rule that defines each number being added.
- S_N — Nth partial sum (dimensionless): The sum of the first N terms: S_N = a₁ + a₂ + … + a_N.
- \lim_{N \to \infty} — Limit as N approaches infinity (—): The value that S_N approaches as we include more and more terms. If this limit exists, the series converges.
Worked Example
Problem. Consider the geometric series ∑_{n=1}^{∞} (1/3)^n = 1/3 + 1/9 + 1/27 + … Determine whether it converges, and if so, find its sum.
Answer. The series converges to 1/2.
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
Cauchy was the first mathematician to place the theory of infinite series on a truly rigorous footing. In his landmark 1821 textbook Cours d'Analyse, he gave the formal definition of the convergence of a series using limits of partial sums — essentially the definition used in every calculus course today. Before Cauchy, mathematicians used infinite series freely but often without rigorous justification, leading to paradoxes. Cauchy established clear criteria for when a series converges, including the Cauchy convergence criterion and several convergence tests. Cauchy was extraordinarily prolific, publishing nearly 800 research papers — so many that the Paris Academy of Sciences had to impose a page limit on submissions specifically because of him.
Related Concepts
- Definition of a Sequence — A series is built directly from a sequence; understanding sequences is the essential prerequisite for defining a series.
- Geometric Series — Geometric series are the most important class of series with a simple closed-form sum, and they illustrate convergence and divergence concretely.
- Convergence Tests for Series — Once a series is defined, convergence tests (ratio test, comparison test, integral test) provide systematic ways to determine whether a series converges.
- Definition of a Limit — The sum of an infinite series is defined as a limit of partial sums; mastery of limits is essential for understanding series rigorously.
- Taylor and Maclaurin Series — Taylor series are one of the most powerful applications of infinite series, expressing functions as infinite polynomial sums.