Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. == Summation == Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction...

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition...

Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula q = e i h = e 2 π i ℏ {\displaystyle q=e^{ih}=e^{2\pi i\hbar...

Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below. == Introduction == The evaluation of nonstandard analysis in the literature has varied greatly. Paul Halmos described it as a technical special development in mathematical logic. Terence Tao summed up the advantage of the hyperreal framework by noting that it allows one to rigorously manipulate things such as "the set of all small numbers", or to rigorously say things like “η1 is smaller than anything that...