Subject: Infinite Geometric Series

Who is asking: Student
Level: Secondary

Question: I ran into a problem when studying how to find the sum of an infinite geometric series. My math book attempts to explain the concept by giving formulas involving sigma and |r|, but it does not really explain how to go about finding the sum of an infinite geometric series. If you could either help me with this or point me in the direction of an informative website that could help me, I'd appreciate it.

Thanks for your time.

Sam Carter:)(:

You have probably already seen the exprression for the sum of a finite geometric series but I want to develope it just to make sure we are using the same notation. Assume that a is positive and the sum of the finite geometric series is.

The usual technique is to multiply both sides by r to get

Subtraction of the second equation from the first gives

or

To see what happens when n increases you need to consider four cases. I am not going to give a rigerous proof but rather try to illustrate the possibilities.

In conclusion, if |r| < 1 then

Otherwise the infinite sum does not exist

The "method" of finding the sum of an infinite geometric series is much more fun than the "formula". The trick is to find a way to have a repeating pattern, and then cancel it out. The classical example is the series

where every term after the first is the previous term divided by 2. You want to find its sum X, so that you have

To find a repetitive pattern, you divide everything by 2 (that is, you do to the whole series what you would normally do just to find the next term).

Distributing the division over the sum, you get

That is, if you start to add up at the second term of your series instead of the first, then you get an answer of X/2 instead of X. Therefore, the first term of the series was already X/2; so 1/2 = X/2, and X = 1. Thus you have

In algebraic notation, you do as follows:

Start with your series:

Divide everything by 2:

Distribute the division:

Subtract the third line from the first:

Or

And then solve for X:

This works also for other series; for instance, if you start at 5, and then add the terms obtained by dividing the previous term by 3, you have

Divide everything by 3:

Distribute:

Subtract the third line from the first:

Or

Then solve for X:

There is a logical and philosophical difficulty with this method: The three little dots represent an infinite continuation. So for instance, in the previous calculation, there are infinitely many subtractions: 5/3 - 5/3 = 0, 5/9 - 5/9 = 0, 5/27 - 5/27 = 0, and so on. Sure, these are all easy, but still you could never write them all down, so you are representing a calculation that you could never carry out completely. This can lead to strange things: If you start at 1 and form a series by MULTIPLYING the terms by 2, you get

Multiply everything by 2:

Distribute:

Subtract the third line from the first:

Or

Solve for X:

Surely this cannot be right! You start at 1, and keep on adding larger and larger positive numbers; how could you end up with a sum of -1? There has to be something wrong with this calculation!

How about the first series we tried:

This looks right: If I start with a pizza, eat half of it on the first day, then half of what is left on the second day, then half of what is left on the third day, and so on, then day by day I am eating

and it seems that over the course of eternity, I should get through my whole pizza.

So, is the calculation correct or not? Difficulties like these stumped scientists for many centuries, and it is reasoning them through that opened the door to calculus and modern mathematics.