Ramanujan Summation for Powers of Triangular and Pronic Numbers

Dr R Sivaraman

doi:10.13005/ojps05.01-02.03

Abstract

The numbers which are sum of first n natural numbers are called Triangular numbers and numbers which are product of two consecutive positive integers are called Pronic numbers. The concept of Ramanujan summation has been dealt by Srinivasa Ramanujan for divergent series of real numbers. In this paper, I will determine the Ramanujan summation for positive integral powers of triangular and Pronic numbers and derive a new compact formula for general case.

Highlights

Definitions and Formulas The sum of first n positive integers is called a triangular number
The first few values of Bernoulli Numbers are given by Theorem 2 The Ramanujan summation for positive integral powers of triangular numbers is given by
The Ramanujan summation abbreviated as RS of is defined by Srinivasa Ramanujan proved a formula connecting Riemann zeta function with Bernoulli numbers

Summary

Introduction

Definitions and Formulas The sum of first n positive integers is called a triangular number. The product of two consecutive positive integers is called a Pronic number and the nth Pronic number is given by Pn= n(n+1) Independent Research Scholar, African Moon University, South West

Results

Conclusion