How can we know the answer to 1-1+1-1+1...?

Question

I was watching this video I noticed that the teacher said that 1-1+1-1+1... equals 1/2. How can we know that? The proof he uses doesn't make sense to me. We go from 1 to 0 to 1 and back again, etc. If it goes on like so forever, where does a fraction come into play?

Answer 1

There are multiple measures of what that comes to but intuitively you might think that the value alternates between 1 and 0, so you could call it a half.

In truth this series never converges on any given number. Depending on how you define addition, the sum to infinity is not properly defined. You might argue that the sum of those numbers is 1 if $\infty$ is an odd number and $0$ if $\infty$ is an even number.But $\infty$ is neither an even nor an odd number.

Summing to infinity cannot truly be evaluated as a sum as it is a process of addition that never ends, rather than a final result. What we can say fairly categorically is that the only acceptable numerical answers are $1, 0,$ and $\frac{1}{2}$.

Answer 2

The series does not converge

But if you treated it as a formal geometric series $1+r+r^2+\ldots= \dfrac{1}{1-r}$ and then let $r=-1$ you would get $\dfrac{1}{2}$

Similarly if $S=1-1+1-1+\cdots$ then you might set up

 S = 1 - 1 + 1 - 1 + ...  
 S =     1 - 1 + 1 - ... 

and adding the two vertically gives

2S = 1

Answer 3

It's a non-convergent series (Grandi's Series), so all we can say is that it diverges. Otherwise you get weird results.

For example:

$$S = 1 - 1 + 1 - 1 + 1 - 1 + ... = 1 - (1 -1) - (1 - 1) - (1-1) - ... = 1$$

or

$$S = 1 - 1 + 1 - 1 + 1 - 1 + ... = (1-1) + (1-1) + (1-1) + ... = 0$$

or

$$1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + ...) = S \implies S = 1/2$$

Answer 4

That series does not converge, because the limit of the partial sums ($0, 1, 0, 1, ...$) does not exist.

Answer 5

For divergent series, you can use Cesàro summation to find your desired limit.

The idea is to take the mean of your partial sums, and we hope it converges:

1 1 1/1
2 0 1/2
3 1 2/3
4 0 2/4
5 1 3/5

etc. you can see the mean goes to $1/2$.