Grandi's series

Grand's Series:

Look at the following series,

S = 1 − 1 + 1 − 1 + ⋯ which can be also represented as

 is called as Grandi's Series. The question that came to the ancient mathematician was, what is the sum of the series?  Well, these mathematician came up with the two approaches as follows:

1. They attack on the series like,

 (1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0.

2. On the other hand some treat it like,

 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1.

Well, both the results were very contradictory. By treating the above series in two different ways, one can get either 0 or 1 as an answer.

In 1703, one Italian mathematician named Guido Grandi came up with the idea that the sum of above series is neither 0 nor 1, but it is 1/2. After which many mathematician of the time disagreed and said, "what? The sum can't be 1/2." This was very hot topic for the debate in the mathematician of that time for some years. After some years, some mathematician researched about the same and were surprised as the Sum 1/2 was making sense. 

One of the approach to find the third value that is 1/2 is as follows:

          S = 1 − 1 + 1 − 1 + ..., so

1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S

1 − S = S

1 = 2S ,so S= 1/2.

Wow! Wasn't that amazing!

Another approach : 

In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists.

For instance, 

The series,

   1+ 1/2 +1/4 +1/16 +1/16 +1/32 +.......upto ♾️, obviously converges to 2.

Now, consider its partial sum, which is the sequence { 1, 3/2, 7/4, 15/8, 63/32,.......}

you can obviously see that the sequence is  converging to the value 2. So, the sum of 1+ 1/2 +1/4 +1/16 +1/16 +1/32 +.......upto ♾️ is 2. 

    Now consider the grand's Series

S = 1 − 1 + 1 − 1 + ... 

The sequence of partial sums of Grandi's series is {1, 0, 1, 0, ...} which clearly does not approach any number. So, grandi's series does not have sum and is divergent. So this process to find the sum of Gradhi's series is not going to work here. But don't worry, we have another way to assign value to the divergent series. Cesaro summation can be used sometimes to assign value to the divergent infinite series. 

Consider the series 1+ 1/2 +1/4 +1/16 +1/16 +1/32 +.......upto infinity whose sequence of partial sum is 

 { 1, 3/2, 7/4, 15/8, 63/32,.......}. You already know this sequence converges to 2. What cesaro summation says, is find the sequence of average of partial sums, and whatever the limit of that sequence is, that is the value of the sum of the series. Means for above sequence average sequence will be { 1/1, (1+3/2)/2, (1+3/2+7/4)/3, (1+3/2+7/4+15/8)/4, ......}

= { 1, 5/4, 17/12,.....} like that. This new sequence also converges to 2. So, it is working.

Let's apply it on Grandi's Series.

S = 1 − 1 + 1 − 1 + ... 

It's sequence of partial sum is, {1, 0, 1, 0, ...} 

Now, averaging it's partial sum gives the sequence { 1/1, 1/2, 2/3, 2/4, 3/5, 3/6,....}

You can actually see the sequence is zoning to 1/2. So, here we go. Now, you can conclude that the sum of grandy series is 1/2.  

Here, we have seen the two methods to find the sum of Grandi's series but the second one was more technical than the first one.

Conclusions:

   1. Grandi's series is divergent series.

   2. But it is summable and some is equals           to 1/2.

Thomson's lamp puzzle is well known philosophical puzzle based on Grandi's series. You can check it on Wikipedia.

Let's have some fun with Grandi's series:  

Consider a person with superpower ability to travel with the speed of light. 

let the person is at A for 1 minute then he travelled with his maximum speed to B and stays there for 1/2 minute. Again he come back to position A and stays there for 1/4 minute. Again he travel to B and stays there for 1/8 minutes. He do this activity infinite number of times.

Now, question is what will be his position exactly after 2 minutes? A or B ?

What if I say he will be present at both A and B at the same time.

What do you think? let me know your opinion in comment section.