We represent a generic sequence as a 1 , a 2 , a 3 ,. .. , and its n-th as a n. In order to define a sequence we must give enough information to find its n-th term. Two ways of doing this are: 1. With a formula. E.g.: a n = 1 n a n = 1 10 n a n = √ 3n − 7 2. With a recursive definition. 1.2. Limit of a Sequence. We say that a sequence a n converges to a limit L if the difference |a n − L| can b...

Here is the abstract CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling curve, surface, solid and object representations;

(1 .2) E(k)ak = Sn,j k=1 k = j (modn!) for n=1, 2,. .. and O~j-n!-1. When sn , j =0 for all n and j, (1 .1) follows automatically. Since every arithmetic progression with modulus m is a disjoint union of (m-1)! arithmetic progressions with modulus m!, Theorem 1 follows from Theorem 2. Also, by using this argument and (1 .2), we see that in Theorem 2 each series Z E (n) an is, n=b(mod m) in fact...

Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie ...

In this paper an example from the history of mathematics is presented and its educational utility is investigated, with reference to pupils aged 16-18 years. Students’ behaviour is examined: we conclude that historical examples are useful in order to improve teaching of infinite series; however their effectiveness must be verified by the teacher using experimental methods, and the primary impor...

In a recent paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! and on the same lines a simple criterion for some fast converging series representing irrational numbers was given. In the present paper, we give some generalizations of our previous results. 1 Irrationality criterion Our considerations in [3] lead us to the following criterion for irrationality, where x...

Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof. Most combinatorial Rogers-Ramanujan type identities (e.g. the Göllnitz-Gordon identities, Gordon’s combinatorial generaliza...

Abstract: An exact solution method in terms of an infinite power series is developed for linear ordinary differential equations with polynomial coefficients. The method is general and applicable to a wide range of equations of any Nth order presented in normal form. The final solution is defined by a linear combination of S functions fj(x) j=1,...,S expressed in the form of a power series, and ...

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, optics, and other fields. In the process studying these topics, very obtain exact solutions equation. this paper, a general Riccati treated as an auxiliary equa...