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A Simple Continuation for Partial Sums

Kamal Saleh

Abstract

In 2014, Ibrahim M Alabdulmohsin wrote a paper called "Summability Calculus" where he developed a method to generalize sigma notation to non-integer upper bounds. His paper included a theorem, known as Theorem 6.1.1 (denoted here as Lemma 2.1 because of its simplicity and location in this paper), but doesn't study it much. Another paper by Mueller and Schleicher also analyzed this formula, but doesn't integrate or differentiate the formula and states some specific applications. This paper will analyze the simple formula that generalizes sigma notation to non-integer upper and lower bounds. We state and prove this formula in Section 2. Section 3 states a few algebraic properties for the sum and product formulae and shows how differentiation of sums and products works. Because integrating a product is challenging, we only analyze the integration of sums in the fourth part of Section 3. In Section 4 we apply the formula in the second section to create analytic continuations for functions defined as partial sums, formulate an infinite series representation to any limit, create a great approximation for functions that approach a certain limit, make an analytic continuation for products, and calculate the sum of anti-derivatives. We then conclude with a discussion of the material of this paper.

A Simple Continuation for Partial Sums

Abstract

In 2014, Ibrahim M Alabdulmohsin wrote a paper called "Summability Calculus" where he developed a method to generalize sigma notation to non-integer upper bounds. His paper included a theorem, known as Theorem 6.1.1 (denoted here as Lemma 2.1 because of its simplicity and location in this paper), but doesn't study it much. Another paper by Mueller and Schleicher also analyzed this formula, but doesn't integrate or differentiate the formula and states some specific applications. This paper will analyze the simple formula that generalizes sigma notation to non-integer upper and lower bounds. We state and prove this formula in Section 2. Section 3 states a few algebraic properties for the sum and product formulae and shows how differentiation of sums and products works. Because integrating a product is challenging, we only analyze the integration of sums in the fourth part of Section 3. In Section 4 we apply the formula in the second section to create analytic continuations for functions defined as partial sums, formulate an infinite series representation to any limit, create a great approximation for functions that approach a certain limit, make an analytic continuation for products, and calculate the sum of anti-derivatives. We then conclude with a discussion of the material of this paper.
Paper Structure (17 sections, 2 theorems, 70 equations, 3 figures)

This paper contains 17 sections, 2 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Simple Result
  3. Base Formula
  4. General Formula
  5. Product Formula
  6. Properties
  7. Fundamental Properties for the Sum and Product
  8. Differentiation
  9. Upper Bound
  10. Lower Bound
  11. PDEs
  12. Integrating Sums
  13. Applications
  14. Analytic Continuations
  15. Function Approximation
  16. ...and 2 more sections

Key Result

Lemma 2.1

If $\lim_{k\rightarrow\infty}f(k)=L$ for some constant $L$ and function $f(x)$ defined on the positive integers, then

Figures (3)

  • Figure 1: Approximation for $1/x$
  • Figure 2: Approximation for $e^{-x}$
  • Figure 3: Approximation for $\frac{\sin x}{x}$

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 4.1
  • proof