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?

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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, the sum of the first 4 terms, and so on).

IF those sums tend towards some certain number (a limit), THEN that is the SUM!

The partial sums for Grandi's series are:

S₂ = 1 − 1 = 0
S₃ = 1 − 1 + 1 = 1
S₄ = 1 − 1 + 1 − 1 = 0
S₅ = 1 − 1 + 1 − 1 + 1 = 1
S₆ = 1 − 1 + 1 − 1 + 1 − 1 = 0
S₇ = 1 − 1 + 1 − 1 + 1 − 1 + 1 = 1
...

As you can see, the partial sums never get anywhere... they form a sequence of alternating 0s and 1s.

That is PROOF that this series does not "converge", or have a definite sum. Instead, it simply diverges.

But, it FOOLS you for a while, doesn't it? It's only when we put the truth of mathematics ON it, that we can see... no, it's not a nice, convergent series with a sum.

It's kind of like, you look at it one way, and you get 0. You look at it another way, and you get 1. You look at it in a few other different ways and get 1/2. It's white, it's black, it's gray... can't tell what it is.

And the series itself "jumps all over the place" with its 1s and negative 1s, never making up its mind. It's like a PRETENDER! Pretending to be a TRUE convergent series... but it is not.

For a comparison, here's a TRUE convergent series (it has a well-defined sum):

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

(It is a simple geometric series)

Its partial sums are:

S₁ = 1
S₂ = 1 + 1/2 = 1 1/2
S₃ = 1 1/2 + 1/4 = 1 3/4
S₄ = 1 3/4 + 1/8 = 1 7/8
S₅ = 1 7/8 + 1/16 = 1 15/16
S₆ = 1 15/16 + 1/32 = 1 31/32
S₇ = 1 31/32 + 1/64 = 1 63/64
S₈ = 1 63/64 + 1/128 = 1 127/128
...

These tend to 2. So, the sum of this series is 2. A true series!