How to refute Grandi's Series

Grandi’s Series is an example of a Cesàro Sum, which gives a sum for a sequence that tends towards infinity. Let’s see how to refute this famous series.

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Grandi’s Series is a very famous series made by an Italian mathematician, philosopher and priest, Guido Grandi. The series in its simplest form can be depicted as:

S = 1–1+1–1+1–1+1–1+…

This series goes on and on and never stops. However, looking at the series, one would say that the answer to the series would be: 0, if the series ends with -1, and 1, if the series ends with +1. But, here is what Grandi did:

Let S = 1–1+1–1+1–1+1–1+… 1–S = 1–(1–1+1–1+1–1+1–1+…) 1–S = 1–1+1–1+1–1+1–1+… (by opening the brackets) 1–S = S (by substituting S from the first step of the derivation) 1 = 2S S = 1/2

Here, Grandi showed that the sum of the series for which we all said was 0 and 1, is actually 1/2! This topic is discussed upon by many mathematicians around the world. There are many more such Cesàro Sums, which you all can read about. But now, I will show how to refute Grandi’s Series.

The first thing that we must do is to look at the final term of the series.

Let us take the last term as –1 S = 1–1+1–1+1–1+1–1+…–1 1–S = 1–(1–1+1–1+1–1+1–1+…–1) 1–S = 1–1+1–1+1–1+1–1+…+1 1–S = S+1 (by substituting S from the first step) 0 = 2S S = 0

Let us take the last term as +1 S = 1–1+1–1+1–1+1–1+…+1 1–S = 1–(1–1+1–1+1–1+1–1+…+1) 1–S = 1–1+1–1+1–1+1–1+…–1 1–S = S–1 (by substituting S from the first step) 2 = 2S S = 1

These are two solutions that we got in at the starting! What did we do different this time? What was different is that Grandi ignored the last term and substituted S irrespective of the last term. By taking the last term into account, we are not finding the Cesàro Sum to the series. This is because, a Cesàro Sum is taken with the limit as infinity. By including the last term, we are indirectly putting a limit to the series. This is why we can use the method to give a value to a limited Grandi’s Series. But with an infinite limit we cannot. Infinty is the cause to many different paradoxes in mathematics. This is also how Srinivasa Ramanujan discovered that the sum of all natural numbers to infinty is -1/12! So remember, the Cesàro Sum is a summation of a series to infinity and cannot be limited.