## Research and Reports on Mathematics

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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: [email protected]

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 , mostly for 0<q<1. Now we prove that for any 0<q<1

## 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 , 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.

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 Sk, 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.

Step 1: Take a unit square S and take rectangles Q1+ with AQ1+ = q and Q2 with AQ2- = 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 , General step k: We repeat the actions of the previous step for Q2k-2 , but we reduce the rectangles by a scale factor of q.

We get that the area of unit square is Hence 