In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.
In this article we prove the Leibniz series for π which states that π4=∑n=0∞(−1)n2⋅n+1. The formalization follows K. Knopp [8], [1] and [6]. Leibniz’s Series for Pi is item 26 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.
In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I-convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I-convergence sense.
We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula with convergence as , in much the same way as the Euler transformation gives with convergence as . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in the Appendix....