### 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

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

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

we get

James also shows TWO WAYS of obtaining the

To get some SANITY into the situation, we need to look at what is true: what is the actual

That has to do with the

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, …