Infinite Series

An infinite series is the sum of an endless sequence of terms. Although such a sum has infinitely many parts, it can still have a finite total when the terms shrink quickly enough; in that case the series is said to converge, otherwise it diverges. The value of a convergent series is defined as the limit of its partial sums, the running totals obtained by adding one term after another. Determining whether and to what a series converges is a central problem in analysis.

Several standard families illustrate the range of possible behaviour. A geometric series, in which each term is a fixed multiple of the previous one, converges precisely when that ratio is smaller than one in magnitude, and its sum has a simple closed form. The harmonic series, formed from the reciprocals of the natural numbers, diverges despite its terms tending to zero, while a general p-series converges only when its exponent exceeds one. Alternating series, whose terms switch sign, converge under the gentler conditions described by Leibniz's theorem. Telescoping series collapse because consecutive terms cancel, leaving only a few surviving pieces.

Power series extend the idea by including a variable raised to increasing powers, allowing functions to be represented as infinite polynomials within a radius of convergence; Taylor and Maclaurin series are important examples. Related combinatorial objects appear throughout this subject, including the binomial coefficients arranged in Pascal's triangle and the Bernoulli numbers, which arise in formulas for sums of powers and in the expansions of certain functions.

Frequently asked questions

When does a geometric series converge?

It converges when the common ratio between successive terms is less than one in absolute value, in which case its sum equals the first term divided by one minus the ratio.

Why does the harmonic series diverge?

Even though its terms approach zero, they do so too slowly; the partial sums grow without bound, so the series has no finite total.

What is a power series used for?

A power series represents a function as an infinite polynomial, which is useful for approximating values, integrating, and analysing functions near a point.