Fun with maths, part 4: 1 + 2 + 3 + 4 + ... = ? The answer will surprise you. (Yes, there is a wikipedia page on this famous infinite series!)

A staple of any Algebra 2 and Pre-Calculus curriculum will include a unit on Sequences and Series. I love teaching this as it allows me the opportunity to go over one of the most famous infinite series, the sum of natural numbers. Although it is clearly divergent in a traditional mathematical sense, by employing the Ramanujan summation we can arrive at a result of:

1 + 2 + 3 + 4 + ... = - 1/12

Surprising no? In so many ways:

  • how does one arrive at a negative value only summing positive numbers?
  • how does one arrive at a fraction only summing natural numbers?
  • how does one arrive at a sum at all?!?

Don't believe me? Here is a copy from Ramanujan's notebook entry:

Another key surprise is that with one little approximation the proof become relatively trivial. In addition, this result is used in physics applications such as string theory. Here we have a graphical representation of such sum:
source: wikipedia

Now onto the simplified proof:

Recall: our goal is to find the sum of the following series S:

S: 1 + 2 + 3 + 4 + ...

Step 1: Let us define the following series:
A: 1 - 1 + 1 - 1  + 1 - 1 ....

Obviously this sum will either equal 0 or 1 depending on if we stop after an even number of terms or odd number of terms. Using the average of the two (also known as using the Cesaro summation) we can say that:

A: 1 - 1 + 1 - 1  + 1 - 1 .... = 1/2

Step 2: Let us define the another series:
B: 1 - 2 + 3 - 4  + 5 - 6 ....

If we add B to itself we get 2B, which happens to equal A! Why? Here it is:
   B: 1 - 2 + 3 - 4  + 5 - 6 ....
+ B:      1 -  2 + 3 - 4  + 5 ...
2B:  1 - 1  + 1  - 1  + 1  -  1 ... = 1/2

Therefore if 2B = 1/2, using simple algebra we arrive at:

B: 1 - 2 + 3 - 4  + 5 - 6 .... = 1/4

Step 3: Let us subtract series B from our original series S (i.e., S - B):

      S:       1 +  2 + 3 + 4 + 5 + 6...

   - B:       1 -  2 + 3 - 4  + 5 - 6 .... 

(we need to flip the signs of the terms of series B, so it becomes - 1 + 2 - 3 + 4 - 5 + 6 ...)

S - B =    0 + 4 + 0 + 8 + 0 + 12 + ... 

Step 4: We can now factor out a 4 from the sum S - B:
S - B =   4( 1 + 2 + 3 + 4 +  ...)

Given that the series in the parenthesis equals our original sum S, and we have a value for sum B (i.e., 1/4) , we can now find the answer to our original question...

Step 5: Solve for S:
 S - B =   4( 1 + 2 + 3 + 4 +  ...)
 S  - 1/4 = 4S      (by substituting our value of B and S)
     - 1/4 = 3S       (by subtracting S from both sides)
     - 1/12 = S        (by dividing both sides by 3)

And there you go. Our proof is now complete:

S: 1 + 2 + 3 + 4 + ... = - 1/12

As the summer continues to roll on, if you are looking for a good film, you could do worse than watching the excellent Man Who Knew Infinity which retells the amazing story of Ramanujan. Enjoy!

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