Extreme Finitism presents: Astounding 1 + 2 + 3 + 4 and so on does not
equal minus one twelfth. This is a response video
to the numberphile video called
astounding 1 + 2 + 3 + 4 and so on
equals minus one twelfth in which it claims if
you add up all the natural numbers all
the way up to infinity the answer is
minus one twelfth in this video we will reveal
the truth about minus one twelfth first off when
we manipulate a series that does not end
the rigorous approach is to work with
respect to the nth term here we have
the series s1 and the S stands for
series not sum, it will be misleading to
call it a sum because the series has
no end point and so cannot have a sum we
have an unspecified number of terms up
to the nth term and the series continues
endlessly after the nth term now we
can add another s1 and shift some terms
and notice how we line up the endless
parts after the nth term here we
simply add these two together then we
divide through by 2 to get this new
expression for series s1 if you use a
less rigorous method you might end up
losing the end part of this result and
be left with s1 equals a half but if you
end up with an endless series being
equal to a fixed value then you've made
a mistake now if we drop the endless
part we find we found an expression for
the sum to the nth term we have derived
the partial sum expression the fixed
part of this expression would be called
the limit if we had a converging series
but with a diverging series like this
one the fixed part is what remains after
this diverging part is removed and it
could be argued that this is what the
informal approach tries to achieve and
so what happens if we use our rigorous
approach to do all the manipulations
that they used in the astounding results
video to get the minus one twelfth
result if we include an expression for
the nth term in each series and then we
do the same manipulations done in the
numberphile video but with rigor then we
get the series s equals minus one twelfth
+ one twelfth + n times n plus 1 over 2 plus
endless terms after the nth term the
informal method dropped these trailing
terms leaving s equals minus one twelfth as the
result but here the first two terms
cancel out leaving an expression for the
partial sum plus endless terms after the
nth term but what about other methods
that get this minus one twelfth result well
these methods are all trying to assign a
value to a diverging series by removing
the diverging parts now we can see from
the partial sum that removing a
diverging part leaves 0 not minus one twelfth
so there is still the question of how
these methods can examine the sum of
positive whole numbers and end up with a
non positive non whole number result
well subtraction operations will bring
negative numbers into play and division
operations will take us out of the realm
of whole numbers now let's see what this
means in terms of a graph here is a
graph of our partial sum function the
graph is plotted for decimals not whole
numbers and includes negative numbers
removing the diverging parts in this
case means removing the shaded end parts
our mysterious value of minus one twelfth is the
limit of the area that remains beneath
the x-axis note that if we take a
function that applies to positive whole
numbers and we examine it with respect
to decimal and negative numbers we will
inevitably introduce a skew of one to
the left or right of the y-axis and we
often get symmetry around x equals a
half or minus a half and our mysterious
value of minus one twelfth is simply a property
of the
way this graph is skewed towards the
negative numbers so let's take negatives
into consideration if we use our partial
sum expression to calculate the sum up
to the fourth term then we get the
correct result of 10 the problem is we
have not defined what should happen if
we use negative values for n the obvious
choice is to aim for symmetry with
symmetry if we plugged minus 4 into our
function we should get a result of minus
10 but as it stands if we plug minus 4
into our function we get plus 6 this is
due to the skew we mentioned earlier to
adjust our function for negative values
we first apply the modulus operator like
so then we multiply by n over the
modulus of N and we get a function that
works in the same way for both negative and
positive numbers if we plot decimal
values for this corrected function we
now see symmetry about the axes and
removing the diverging parts will no
longer leave anything resembling minus
one twelfth and what about the zeta function at
minus 1 well if we follow the same logic
to adjust for negative numbers as we did
earlier we get symmetry about the axes
and the function would not be defined at
minus 1 so how astounding is all of this
well 1 + 2 + 3 and so on does not equal
minus one twelfth and minus one twelfth is not some
mysterious value the Infinity has
revealed to us it is merely a property
of the skew we get when we neglect to a
adjust for negative values so maybe this
is not such an astounding result after
all this video was brought to you by
Extreme Finitism dot com in association
with Reveal Computer Software