Infinite series, the tests that decide when they converge, and the Taylor expansion that turns functions into polynomials are among the most useful constructions in all of applied mathematics. A series is simply a limit of partial sums, and the first question is always whether the limit exists and what it equals. The geometric series, whose partial sums have a closed form, converges exactly when the common ratio has magnitude strictly less than one, and its behaviour at and beyond the boundary is instructive: at a ratio of negative one the partial sums oscillate without settling, and at a ratio of one they grow without bound. The harmonic series, whose terms decrease to zero, still diverges, and the surprise of that fact in the eighteenth century caused no lasting harm; the p-series generalises it, converging precisely when the exponent exceeds one, a criterion that supplies an immediate test for a large class of series. Comparison and limit comparison then handle series that resemble known ones, the ratio test, due to d'Alembert, settles the common case in which successive terms eventually have a constant ratio, the root test, associated with Cauchy, catches series with ratios that do not exist, and the alternating series criterion of Leibniz, which requires only decreasing terms tending to zero, delivers conclusions such as the expansion of the logarithm of two as a slowly convergent sum. The decisive distinction is between absolute and conditional convergence, that is, between series whose series of absolute values converges and those for which it does not. Absolute convergence permits reordering, permits term-by-term differentiation and integration, and permits taking limits term by term, whereas conditionally convergent series can be made to sum to different values by reordering their terms, a fact demonstrated by Riemann, so that any manipulation ignoring the distinction is a genuine error rather than a technicality. The property that legitimises those manipulations is uniform convergence, which demands that the tails of the partial sums are small everywhere at once rather than merely at each point in its own way. The criterion that guarantees it most often is the Weierstrass comparison test, in which the terms of a series of functions are dominated by the terms of a convergent numerical series, and the two celebrated theorems that rest on it are the interchange of a limit with a sum and the continuity and term-by-term differentiability of a uniformly convergent series of continuous functions. Power series have their own theory: a power series converges absolutely inside some radius, may or may not converge at the two endpoints, and the radius is found from the ratio of successive coefficients, with the whole interval of convergence forming a single connected region. Taylor's theorem then says that a sufficiently smooth function at a point agrees there with a polynomial built from its derivatives, up to an error that can be bounded explicitly by the next derivative times a power of the distance, and this remainder estimate is what makes the expansion genuinely useful rather than merely suggestive. Maclaurin series for the exponential, the trigonometric functions, the logarithm, the arctangent and the binomial expression supply a numerical library with a great variety of behaviour, some converging in a few terms and some, such as the naive series for pi, converging so slowly that practical calculations replaced it with formulas built from rapidly convergent arctangents. In physics the same expansions appear wherever a quantity is close to a regime in which it simplifies: small oscillations are described to leading and next order in the amplitude, orbital perturbation theory expands a difficult motion in a small parameter, and the method of multiple scales separates a fast oscillation from a slow envelope. In numerical analysis the same ideas underpin truncation error estimates, the derivation of Runge-Kutta methods, and the Euler-Maclaurin formula that connects sums to integrals and yields Stirling's approximation to factorials, so the theory of series is far from an academic exercise in manipulation. Every one of these results is conditioned on convergence, and the most common error in practice is a manipulation performed term by term on a series that converges only conditionally, where the result may fail to converge altogether or may converge to a different value. The remedy is to establish absolute convergence first, which is usually easy and costs nothing. It is also worth remembering that a series representation is a choice rather than a discovery, since the same function has one expansion about each centre with a different radius of validity, and choosing the centre nearest the point of interest is what makes the truncation error small. Numerical tables exploit exactly this fact, because values of logarithms, trigonometric functions and factorials are computed from such expansions with an attached error estimate, and the great advantage of such an estimate is that it lets a caller decide how many terms are actually worth the effort of evaluating.
