Finding the fallacy in this broken proof

Today, a friend gave me a "proof" of $1=2$ and challenged me to find the fallacy.

$1 = 1$

$1 = 1 + 0 + 0 + 0 ...$

$1 = 1 + 1 - 1 + 1 - 1 + 1 - 1 ...$

$1 = 2 - 1 + 1 - 1 + 1 - 1 ...$

$1 = 2 + 0 + 0 ...$

$1 = 2$

My answer was that once you turn the initial $1 + 1$ into a 2, everything is offset so a $-1$ is always left at the end no matter how many times it is repeated. This negative one balances out the $2$ at the beginning so $1=1$ still holds true. I.e.

$$1 = 1 + 0 + 0 + 0 ... = 1 + (1 - 1) + (1 - 1) + (1 - 1) = 2 + (-1 + 1) + (-1 + 1) - 1$$

However, my friend claimed that my answer only applies if the $+ 1 - 1$ repeats for a finite number of times. He argues that because the sequence repeats infinitely and things work differently when working with infinity, my answer is not valid.

Can anyone enlighten me to the true fallacy in this proof?

Solution 1:

Some users pointed me to read up on converging and diverging series.

As I currently understand it, equating $1+0+0+0... = 1+1−1+1−1+1−1...$ is the fallacy because the series on the right does not converge (much less to 1) - therefore, they are not equal.

To prove that this is the fallacy, we can use convergent tests to show that the two sides of the equation are not equal.

Am I correct in my deduction?

Solution 2:

We need to discuss two related things here: what mathematicians (as opposed to the general public) mean by "$\dots$", and how we produce a way of working with infinite sums that both makes sense, and is consistent.

What does "$\dots$" mean?

When the layperson writes "$\dots$", they normally mean either

"and so on, until the obvious logical endpoint", or
"and so on, forever"

But mathematicians need something more precise:

The first usage is written mathematically by writing the endpoint afterwards: $a_1+a_2+\dotsb+a_n$ simply means to add all the numbers in the sequence with labels between $1$ and $n$. Provided we know what all these are, this is straightforward: for example, $ 1+2+\dotsb+100 = 5050 $.
The second usage is a problem: $a_1+a_2+\dotsb$ has no last member, so we need to add infinitely many things.

How do we add infinitely many things?

The answer is that we don't. This sounds silly, but first ask yourself this: how do you add three things together? It's impossible to add three numbers at once, so what you do is add two, and then add the third to the result. Similarly for $n$ things.

What about infinitely many? The clue lies in the previous paragraph: we compute $$\begin{align} s_1 &= a_1 \\ s_2 &= a_1 + a_2 = s_1 + a_2 \\ s_3 &= a_1 + a_2 + a_3, \end{align} $$ and so on, using the rule that $s_n = s_{n-1} + a_n$. $s_n$ are called the partial sums.

But this is still a load of finite sums. How do we assign a value to the whole infinite sum?

The Limit of the Partial Sums

We say that $a_1+a_2+\dotsb$ has a sum (or is summable) if $s_n$ tends to a limit, say $s$, as $n$ tends to infinity. In particular, we can make $s_n$ as close to $s$ as we like by taking $n$ large enough. In particular, it is therefore required that the terms in the series converge to zero, since $s_{n}-s_{n-1}=a_n$. This is obviously not the case for Grandi's series.

The major advantages of this definition include that it assigns a unique sum to $a_1+a_2+\dotsb$,

$ ka_1 + ka_2 + \dotsb = ks $, and
if $a_n'=a_{n+1}$, $a'_1 + a'_2 + a'_3 + \dotsb = s-a_1 $.

How, therefore, does one manipulate these objects that look like series, but do not have a sum?

Assigning values to divergent series

We would like a method to give divergent series a consistent meaning so that we can manipulate them in a meaningful way: i.e., so we find unique values, compatible with previous theory, and that we can apply simple algebraic operations to. We write $S(a_n)$ to be the value assigned to $a_1+a_2+\dotsb$ We normally therefore ask that a summation method has

Scaling: $ S(ka_n) = kS(a_n) $,
Addition: $ S(a_n+b_n) = S(a_n)+S(b_n) $,
Relabeling: $a_1+S(a_{n+1}) = S(a_n)$ (not the standard name, but never mind),
Regularity: If $a_1+a_2+\dotsb$ is convergent, $S(a_n)$ agrees with the conventional sum,

and I think most people would agree that all of these are sensible things you want a summation method to do.

Supposing that we demand that Properties 1. and 3. hold for our summation technique. Then Grandi's series satisfies $$ s = 1-1+1-1+\dotsb = 1-(1-1+1-1+\dotsb) = 1-s, $$ so any "reasonable" method of summation gives Grandi's series the value $s=1/2$.

For much more information on this, I recommend G.H. Hardy's definitiv work on the subject, Divergent Series. In particular, Chapter 1 discusses this and other series, and the four properties given above are in § 1.3.