Convergence of Parametrized Series

Description

This lecture explores the convergence of a series parameterized by alpha, focusing on finding the values of alpha for convergence. By comparing the series to the zeta function, it is shown that alpha must be strictly positive for convergence. The general term is studied by comparing it to a known series, leading to the conclusion that the two series are of the same nature. By solving an inequality involving alpha, it is determined that the series converges when alpha is positive and less than or equal to 0.5. The lecture concludes with a discussion on finding the roots of a polynomial to determine convergence.

Official source

https://mediaspace.epfl.ch/media/0_4lhlsn2f

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Related concepts (22)

Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.

Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. The Hurwitz zeta function has an integral representation for and (This integral can be viewed as a Mellin transform.

Lerch zeta function

In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. The Lerch zeta function is given by A related function, the Lerch transcendent, is given by The transcendent only converges for any real number , where: or and .

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K.

Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.