A never Ending Contradiction

Things can get weird when we deal with infinity. Consider the following sum:

Grandi

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

It’s called Grandi’s series, after Italian mathematician, philosopher, and

priest Guido Grandi (1671-1742).

It is weird but grouping the series in different ways gives different results.

Let’s look at the first way,

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

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

S = 0

Here, it is easy to say that the answer is 0 as each individual bracket is equal to 0.

There’s another way of solving this,

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

S = 1 + (- 1) + 1 + (-1) + 1 + (-1) + 1 + (-1) + 1 …

S = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + …

S = 1

Now, we have an answer which is equal to 1.

Again,

What if we subtract 1 from both sides of the equation? it is obvious that it will remain the same. Or will it?

Let’s find out,

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

1 - S = 1 - ( 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 …)

1 - S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 …

1 - S = S

2S = 1

S = ½

We’ve got S = ½

So, what is the actual answer to this series? Actually, no one knows and it is quite controversial because one can reach towards 2 conclusions,

The series 1 − 1 + 1 − 1 + ... has no sum.
The sum is 1/2 .

Infact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts.

After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.

-Niraaj Ojha (niraaj.ojha@gmail.com)

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