Geometric Proof of the Sum of Geometric Series

Korus P

doi:10 .4172/2327-4581.1000411

Short Communication, Res Rep Math Vol: 2 Issue: 3

Geometric Proof of the Sum of Geometric Series

Korus P^*

Department of Mathematics, Juhasz Gyula Faculty of Education University of Szeged, Hungary

*Corresponding Author : Korus P
Department of Mathematics, Juhasz Gyula Faculty of Education University of Szeged, Hattyas utca 10, H-6725 Szeged, Hungary
E-mail: korpet@jgypk.u-szeged.hu

Received: March 28, 2018 Accepted: July 10, 2018 Published: September 8, 2018

Citation: Korus P (2018) Geometric Proof of the Sum of Geometric Series. Res Rep Math 2:3

Abstract

The well-known formula for the sum of the geometric series is For arbitrary -1<q<1. Among analytic proofs, geometric proofs were also given for this formula, see [1], mostly for 0<q<1. Now we prove that for any 0<q<1

Keywords: Algebra, Applied Mathematics, Arithmetic, Calculus, Combinatorics, Computational Mathematics, Geometry & Topology

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Introduction

The well-known formula for the sum of the geometric series is

For arbitrary -1<q<1. Among analytic proofs, geometric proofs were also given for this formula, see [1], mostly for 0<q<1. Now we prove that for any 0<q<1

holds in the ‘Positive case’ and

in the ‘Alternating case’.

Positive case

As in Figure 1, we do the following process.

Figure 1: Positive case.

Step 1: Take a unit square S₁ and take rectangles Q₁ with area = q and Q₂ with = q₂. We also take ‘adjunct’ rectangles R₁ with and R₂ with

We get a remaining square S₂ of side length q.

Step 2: We repeat the actions of the previous step for S₂, but we reduce the rectangles by a scale factor of q. Then we get Q₃; Q₄; R₃; R₄ with and a remaining square S₃ of side length q₂.

General step k: We repeat the actions of the previous step for S_k, but we reduce the rectangles by a scale factor of q.

We get that the area of unit square S₁ is

Hence

Alternating case

As in Figure 2, we do the following process.

Figure 2: Alternating case.

Step 1: Take a unit square S and take rectangles Q₁⁺ with A_Q₁⁺ = q and Q₂⁻ with A_Q₂^- = q². We also take ‘adjunct’ square R₁⁺ of side length 1 with and rectangle R₁^- of side lengths q and 1 with (Plus and minus signs indicate the signs of the areas of rectangles within the final sum.)

Step 2: We repeat the actions of the previous step for square Q₂⁻ of side length q, but we reduce the rectangles by a scale factor of q. Then we get with ,