Series (mathematics)

The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

[2][3] Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola.

[4][5] The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton.

[6] The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy,[7] among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

This reasoning was applied in Oresme's proof of the divergence of the harmonic series,[28] and it is the basis for the general Cauchy condensation test.

Using comparisons to flattened-out versions of a series leads to Cauchy's condensation test:[29][30] if the sequence of terms

The most general methods for summing a divergent series are non-constructive and concern Banach limits.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent.

When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.

While many uses of power series refer to their sums, it is also possible to treat power series as formal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition.

Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions.

In the most common setting, the terms come from a commutative ring, so that the formal power series can be added term-by-term and multiplied via the Cauchy product.

Like the zeta function, Dirichlet series in general play an important role in analytic number theory.

In general they do not converge, but they are still useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms.

Infinite series play an important role in modern analysis of Ancient Greek philosophy of motion, particularly in Zeno's paradoxes.

The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.

However, in modern philosophy of motion the physical side of the problem remains open, with both philosophers and physicists doubting, like Zeno, that spatial motions are infinitely divisible: hypothetical reconciliations of quantum mechanics and general relativity in theories of quantum gravity often introduce quantizations of spacetime at the Planck scale.

[78][79] Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today.

He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series,[5] and gave a remarkably accurate approximation of π.

Cauchy advanced the theory of power series by his expansion of a complex function in such a form.

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of

Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt (1853).

General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others.

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847–48).

Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula.

Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem.

Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and du Bois-Reymond.

Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.

[83] This generalization introduces two main differences from the usual notion of series: first, there may be no specific order given on the set