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 [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

## 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.

**Step 1**: Take a unit square *S _{1}* and take rectangles

*Q*with area =

_{1}*q*and

*Q*with =

_{2}*q*. We also take ‘adjunct’ rectangles

_{2}*R*with and

_{1}*R*with

_{2}We get a remaining square *S _{2}* of side length q.

**Step 2:** We repeat the actions of the previous step for *S _{2}*, but we reduce the rectangles by a scale factor of

*q*. Then we get

*Q*;

_{3}; Q_{4}; R_{3}*R*with and a remaining square

_{4}*S*of side length

_{3}*q*.

_{2}**General step k**: We repeat the actions of the previous step for

*S*, but we reduce the rectangles by a scale factor of

_{k}*q*.

We get that the area of unit square *S _{1}* is

Hence

**Alternating case**

As in **Figure 2**, we do the following process.

**Step 1:** Take a unit square S and take rectangles *Q*_{1}^{+} with *A*_{Q1+} = *q* and *Q _{2}*

^{−}with

*A*

_{Q2-}=

*q*. We also take ‘adjunct’ square

^{2}*R*

_{1}

^{+}of side length 1 with and rectangle

*R*

_{1}

^{-}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 *Q _{2}*

^{−}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

*Q*

_{2k-2}^{−}, but we reduce the rectangles by a scale factor of

*q*.

We get that the area of unit square is

Hence

## References

- Nelsen RB (1993) Proofs without words: Exercises in visual thinking. MAA.