Can an Endless String of Numbers have a Sum?

If you tutor math 12, you’ll have come across the premise

1 + 1/2 + 1/4 + 1/8 +….

Before we go any further, a reminder on how to multiply fractions:

(2/3) x (4/5) = 8/15.  You just multiply the two top numbers and put that number on top.  Then you multiply the two bottom ones and put that number on the bottom.

Now, back to 1 + 1/2 + 1/4 + 1/8 + 1/16+….(the terms continue forever)

To generate the next term, you just multiply the previous one by, in this case, 1/2.

(1/4) x (1/2) = 1/8

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +….

In spite of the infinite number of terms, the answer is 2.

1 + 1/2 + 1/4 + 1/8 + 1/16 +1/32 + 1/64 + (the terms go on forever)….= 2.

How can there be a sum of an unending number of terms?  Well, there is a proof for it – but that doesn’t mean you have to believe it.  Perhaps the best answer is that from a mathematical point of view, you can add up an infinite number of terms.  However, in this context, two things have to be true:

1) You must always multiply the current term by the same factor to get the next one.

2) That multiplying factor must be less than 1.

So there isn’t a sum to 1+1/2+1/3+1/4 + (the terms continue forever)….because the next number is not a constant multiple of the one before.

There isn’t a sum to 1 + 1 + 1 + (the terms continue forever)….because we’re multiplying the current term by exactly 1 to get the next term.  We must multiply by the same number each time, but it must also be less than one.

There is a sum to 1 + 2/3 + 4/9 + 8/27 +….(infinite terms).  Note that we are always multiplying by 2/3 to get the next term.  The sum is 3.

The formula for the sum is

S=a/(1-r), where

a is the first term

r is the multiplying factor to get the next term (called the common ratio)

If you use a calculator, be sure to put the brackets around the denominator as shown in the formula.

Thanks for stopping by.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.