What is the Sum of the Sequence 1, 1/2, 1/4, 1/8, 1/16, ...?
A pictorial representation of 1 + 1/2 + 1/4 + 1/8 + ...
A mathematical joke
An infinite number of mathematicians walk into a bar. The first mathematician orders a pint of beer, the second mathematician orders half a pint, the third mathematician orders a quarter of a pint, and so on with each mathematician ordering half of the previous order. The barman looks at the mathematicians, shakes his head and proceeds to pour two pints. He then places them down on the bar and says 'You mathematicians. You should know your limits!'
What is happening in the joke?
The mathematicians total order is the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 + 1/16 +... where each number is half of the one before it. At first glance it seems like this shouldn't have an answer. We are adding an infinite number of positive fractions together, so you would think that the total would keep increasing to infinity. The barman in the joke obviously thinks otherwise. So who is right?
We are going to look at two ways of reaching the answer; an algebraic method and a pictorial method.
We are going to look at two ways of reaching the answer; an algebraic method and a pictorial method.
The algebraic method
Our series 1, 1/2, 1/4, 1/8, … is known as a geometric sequence as we use a common multiplier to get from one term to the next. In this case the multiplier is 1/2. If we multiply the first term 1 by 1/2, we get the second term, 1/2. If we multiply this second term 1/2 by 1/2, we get the third term 1/4 and so on.
To find the sum of a geometric sequence we can use the following formula:
To find the sum of a geometric sequence we can use the following formula:
Sum to infinity = a /(1 - r)
In this formula 'a' represents the first term in the sequence and 'r' represents the common ratio (the multiplier).
To see how we derive this formula, you can read this article on 'How to Find the Sum of a Geometric Sequence'.
With our sequence, a = 1 and r = 1/2, so substituting these into our formula gives:
S∞ = 1 / (1 − 1/2) = 1 / (1/2) = 2
We've come to the surprising result that this infinite sum of positive numbers has a limit, 2. It doesn't matter how many terms of the sequence we add together, the total sum cannot exceed 2. In fact, it will never quite reach 2. The sum of the numbers in this sequence will get closer and closer to 2 (infinitely close), but never quite reach or exceed it. We call this the limit of the series.
Mathematically we get:
limit n→∞ ∑ 1/2^n = 1 + 1/2 + 1/4 + 1/8 + … = 2 (for n starting from 0).
To see how we derive this formula, you can read this article on 'How to Find the Sum of a Geometric Sequence'.
With our sequence, a = 1 and r = 1/2, so substituting these into our formula gives:
S∞ = 1 / (1 − 1/2) = 1 / (1/2) = 2
We've come to the surprising result that this infinite sum of positive numbers has a limit, 2. It doesn't matter how many terms of the sequence we add together, the total sum cannot exceed 2. In fact, it will never quite reach 2. The sum of the numbers in this sequence will get closer and closer to 2 (infinitely close), but never quite reach or exceed it. We call this the limit of the series.
Mathematically we get:
limit n→∞ ∑ 1/2^n = 1 + 1/2 + 1/4 + 1/8 + … = 2 (for n starting from 0).
The pictorial method
While the algebraic method is great and often enough to convince people of this strange result, the pictorial method is a helpful tool in picturing what is actually happening and visualising our sum's inability to reach the limit of 2.
To start with, imagine you have two sheets of paper.
To start with, imagine you have two sheets of paper.
Two sheets of paper
Now we have our sheets of paper we are going to start colouring them in by following the sequence. The first two terms are 1 and 1/2, so we will colour in 1 whole sheet and 1/2 of the other sheet.
1 + 1/2 on paper
The third term in the sequence is 1/4. We can add this to our diagram by colouring in 1/4 of the second rectangle.
1 + 1/2 + 1/4
Let's skip a few steps now and colour in 1/8 and 1/16 of our second rectangle.
1 + 1/2 + 1/4 + 1/8 + 1/16
We can already see from the the pictures we have so far that each time we colour in the next fraction of the sequence, we are colouring in half of the remaining space.
For the next step, we want to colour in 1/32. We can see that the remaining white space is 1/16 and so our 1/32 is half of this. When we then colour in 1/64, we are again colouring in half of the remaining space.
As each step is only colouring half of the remaining space, it will always leave the other half uncoloured. Therefore there will always be an uncoloured part left, no matter how far along the sequence we get, and the second sheet will never be completely coloured in. The uncoloured part is why the sum will always remain smaller than 2, and because the uncoloured parts become infinitely small, we can see that the sum will get infinitely close to 2.
For the next step, we want to colour in 1/32. We can see that the remaining white space is 1/16 and so our 1/32 is half of this. When we then colour in 1/64, we are again colouring in half of the remaining space.
As each step is only colouring half of the remaining space, it will always leave the other half uncoloured. Therefore there will always be an uncoloured part left, no matter how far along the sequence we get, and the second sheet will never be completely coloured in. The uncoloured part is why the sum will always remain smaller than 2, and because the uncoloured parts become infinitely small, we can see that the sum will get infinitely close to 2.
Summary
So we have discovered that the barman was correct. The limit of the infinite series 1 + 1/2 + 1/4 + 1/8 + … does indeed equal 2, hence the total order of the mathematicians was two pints.
Comments
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