Infinite Series: My First Introduction

A BLAST FROM THE PAST…

There I was, 14 years old after an average day of school,nothing to look forward to except the madness that the next day would bring with my friends, my mother’s dinner and a bit of the daily gaming,  I would always do homework when I had to, if I had to. I would always leave it till the last day because I really couldn’t bother, especially for Maths class.

I would sit down, learn about basic algebra, solving polynomials, probability trees and other items of that sort. I would be good at them because they were not particularly challenging and as a result, I would go bad at them only because I couldn’t bother to put in the work. Everything just seemed…basic.

“Is this all Maths had to give?” I would say to myself.

Were you to ask me then, what was the point of it all, what were the uses of the Mathematics I was learning or what else lay beyond what we were learning at secondary school (or middle school for my American viewers), I would not have been able to tell you.

Then, while surfing the internet and YouTube, I stumbled across this oddity…

\frac{1}{2}+\frac{1}{4} + \frac{1}{8} +\cdots = 1

This immediately appeared strange. I had never seen nor even come across anything like this before, or anything close to it for that matter…the left hand side was adding fractions forever…

“I’ve never added things forever…” I thought to myself. Nothing I knew could quite explain the rationale for this anomaly to be true.

I immediately continued researching and learnt a bit more on what it meant. The process went as follows:

  1. Grab the number \frac{1}{2}
  2. Divide it by 2
  3. Add it to the previous result
  4. Repeat Steps 2 and 3 infinitely often

In other words, what we were doing was the following…

 \frac{1}{2} = 0.5

\frac{1}{2}+\frac{1}{4} = 0.75

\frac{1}{2}+\frac{1}{4} + \frac{1}{8} = 0.875

\frac{1}{2}+\frac{1}{4} + \frac{1}{8} + \frac{1}{16} = 0.9375

\cdots

The result got closer and closer to 1, but never actually got to it. It immediately made sense as to why, after an infinite number of steps, it should be not “kind of close to 1”, but exactly equal to 1.

ezgif.com-gif-maker.gif

As the above animation shows, if we think of the entire area of the pie to be 1, then if you consecutively add \frac{1}{2}, \frac{1}{4},\frac{1}{8}  , then you progressively get closer and closer to 1. Continuing this process to infinity, we obtain exactly 1.

I remember showing my friend this rather amazing fact, and he quite rightly had a problem with it stating that it can’t possible exactly equal 1, since at every step what we’re adding is always going to produce a number that is less than 1.

This stumped me up for quite a while, because he was right! After every step, it always remains less than 1.

0.5, 0.75, 0.875, 0.9375,\cdots

It was only until I gained a more formal knowledge of Mathematics that I learnt the concept of a LIMIT, and the rather amazing fact that things that hold for finite cases do not necessarily hold up to infinity. 

There is also a very neat proof of the fact that this infinite sum is equal to 1, that I might or might not known about!

Suppose that our infinite sum is equal to some number S i.e. S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\cdots .

Now multiply S by 2 to obtain:

\begin{array}{rcl} 2S & = & 1 + \frac{1}{2} + \frac{1}{4} + \cdots \\ 2S -1 & = & \frac{1}{2} + \frac{1}{4} + \cdots \end{array}

But the right hand side is just S !

This means that we obtain:

\begin{array}{rcl} 2S-1 &=& S \end{array}

and bringing S subject of the formula, we obtain that S=1 .

Therefore,

 S= \frac{1}{2} + \frac{1}{4} + \frac{1}{8}+\cdots = 1


I didn’t really feel very much more interested after learning about this problem, even though it was my first proper introduction to the weird and wonderful side of Mathematics that people rarely ever see before University.

Who knows, maybe it was my friend who convinced me that such a thing was maybe less mysterious than it actually was, and more was just a rash estimation…or it could have been the fact that I was still stuck in a place which didn’t really spark my creative interest into such questions. I distinctly remember learning about functions like f(x)=x+1  and thinking to myself “These seem so useless…”. Oh how wrong I was.

Regardless, to have had that introduction to something which later on would become a very big part of my life is a very intimate moment to have. Rare are those occasions when something brings out that child like nature within you, a reminder of those times like that you’d have before. I’ve recently experienced that again, now. It has made me think of this very first experience and how Mathematics really does never cease to amaze me, time and time again.

Thanks for reading!

-Sinthorel

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