Several methods return this -1/12 result. Obviously no method can add up infinitely many non-zero terms, and so what they do is produce is a limit value. This limit value is merely the limit of the area between the x-axis and the partial sum formula (from x = -1 to 0).
Just as -1/12 is supposedly the sum of all natural numbers, the 'sum of the squares of natural numbers' (and indeed any even power) is supposedly zero, and the 'sum of the cubes of the natural numbers' is supposedly equal to or related to 1/120. Again these values are the simply the limit of the area of the respective partial sum expressions between -1 and 0.
If we plot the respective partial sum function, we see the increasing area under the curve in the positive direction (from x = 0) appears to be equal-and-opposite-to the increasing area under the curve in the negative direction (from x = -1). If we claim these areas cancel each other out, we are left with the area between -1 and 0.
In the case of 1+2+3+4+..., the partial sum S = n(n+1)/2, and the limit of the area for the region between 0 and -1 is the definite integral of this function from -1 to zero, which is -1/12.
Note that when we take a function that applies to positive whole numbers, and we plot it for decimal and negative values, the change from whole numbers to decimals often produces a region of length 1 to the left or right of the y-axis with symmetry around the line x = -0.5 or x = 0.5. However, if we said to ourselves "before we plot this function for negative values, let’s make it work for negative values just the same as it works for positive values" then we would no longer get results like -1/12.
For example, if we wanted our partial sum expression n(n+1)/2 to work for the sum of negative whole numbers in the same way it works for positive whole numbers then we replace n by the modulus of n and multiply the whole thing by n/(modulus n), giving
This adjusted function does not produce the -1/12 area between 0 and -1 when plotted.
In summary, this -1/12 result is simply one huge blunder. The mistake is one of taking a function that applies to just positive whole numbers, manipulating it in ways that bring fractions and negative numbers into play (such as by using division and subtraction operations), and then interpreting the result as though it still relates to positive whole numbers. There is nothing weird or mysterious going on, except for the delusion caused by the strong desire to have a way of adding up 'infinitely many' terms.
This is very similar to the muddled argument of the so-called 'proof by geometric sum formula' that 0.999… equals 1. If we define 0.999… to be equal to the limit of the series 9/10 + 9/100 + … then we can find the limit of this series but we cannot 'prove' that 0.999… equals this limit; it is merely claimed to be true by definition (in which case proof is irrelevant because we only define things that cannot be proven).
The geometric sum formula is a formula for finding the limit of a geometric series; it does not add up infinitely many terms. By calling this the 'geometric sum formula' instead of the 'limit of a geometric series formula' it creates the illusion that we have a formula for adding up infinitely many terms and thus this constitutes a 'proof ' for 0.999… = 1.
There are no formulas or methods that can add up 'infinitely many' terms.
The main flaw in the method that manipulates endless series (used in the Numberphile video) is that we must work with respect to the nth term in BOTH series. We cannot use n terms from one series and n+1 terms (or any number of terms) from another in order to make the trailing parts cancel out.
You may ask why we can’t use the trailing parts in this flexible way. The answer is because it leads to contradictions. For example, if we start with S = 1 + 1 + 1 +…, then we can evaluate S – S to be any value we like by matching up the two series at different starting points. Placing S – S on both sides of an equals sign leads to 1=0, 1=2 or anything else you choose.
For more information about why the notion of 'infinity' should be removed from mathematics go to ExtremeFinitism.com.