### The series of plus 1, minus 1, plus 1, minus 1

I greatly enjoyed watching the movie below, by Dr. James Grime, also known as the numberphile (thanks, hubby!).

He takes the infinite sum (series) of alternating ones and minus ones:

1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 ...

Does this series have a SUM?

Curiously, if we place parenthesis into it this way:

(1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + ...

we clearly get ZERO as the sum.

BUT if we place parenthesis into it in a different way:

1 + (− 1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + 1) + ...

we get ONE as the sum!

WHAT IS GOING ON?

James also shows TWO WAYS of obtaining the sum of 1/2 for this series (see the video and wikipedia page on Grandi's series).

CAN YOU KEEP YOUR HEAD ON STRAIGHT ANYMORE?

To get some SANITY into the situation, we need to look at what is true: what is the actual DEFINITION for a SUM of an infinite series?

That has to do with the partial sums. We need to look at the partial sums (that is, the sum of the first 2 terms, the sum of the first 3 terms, …

He takes the infinite sum (series) of alternating ones and minus ones:

1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 ...

Does this series have a SUM?

Curiously, if we place parenthesis into it this way:

(1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + ...

we clearly get ZERO as the sum.

BUT if we place parenthesis into it in a different way:

1 + (− 1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + 1) + ...

we get ONE as the sum!

WHAT IS GOING ON?

James also shows TWO WAYS of obtaining the sum of 1/2 for this series (see the video and wikipedia page on Grandi's series).

CAN YOU KEEP YOUR HEAD ON STRAIGHT ANYMORE?

To get some SANITY into the situation, we need to look at what is true: what is the actual DEFINITION for a SUM of an infinite series?

That has to do with the partial sums. We need to look at the partial sums (that is, the sum of the first 2 terms, the sum of the first 3 terms, …