The sum of all natural numbers: 1 + 2 + 3 + 4 + .... Part 2

Many people know that

1+2+3+ dots=βˆ’ dfrac112.

But in reality

1+2+3+ dots=βˆ’ dfrac18.

Let us consider in more detail the first result. Of course, a series of natural numbers diverges in the classical sense (in the sense of convergence of a sequence of partial sums: it, of course, has no limit). In this article, the author mentions other summation methods, such as the Cesaro method and the Abel method. Here are some examples: the sum of such a series

 sum limitsn geqslant0(βˆ’1)n=1βˆ’1+1βˆ’1+1βˆ’1+ dots

using the cesaro method will be equal  dfrac12.

Another example:

1βˆ’2+3βˆ’4+5+ dots= dfrac14.

In my opinion, it is wrong to say that the sum of the first row is equal to  dfrac12; correctly say that the sum of the first row in the sense of Cesaro is equal to  dfrac12. Similarly for the second: its sum in the sense of Abel is equal to  dfrac14.

In view of this, in the first result (that βˆ’ dfrac112) there is a substitution of concepts, which leads to a contradiction with common sense.

We now consider in more detail the second result. First, we denote the entire amount for X:

1+2+3+4+ dots=X.

Now we perform the following transformations:

1+2+3+4+ dots=1+ underbrace2+3+49+ underbrace5+6+718+ underbrace8+9+1027+ dots=

1+9+18+27+ dots=1+9 underbrace left(1+2+3+ dots right)X=X.

From here

1+9X=X RightarrowX=βˆ’ dfrac18.

There is another solution. Combine the terms in another way:

1+2+ underbrace3+4+5+6+725+ underbrace8+9+10+11+1250+ dots=

=1+2+25 underbrace left(1+2+3+ dots right)X=X,


1+2+25X=X RightarrowX=βˆ’ dfrac324=βˆ’ dfrac18.

In fact, starting from the top three, we can distinguish 7 terms, the sum of which will be 49, and we will come to the equation


which will give the same result.

In general, you need to act like this: select the first nterms, and then in parentheses take 2n+1terms:

1+ dots+n+ underbrace left(n+1+ dots+3n+1 right)(2n+1)2+ underbrace left(3n+2+ dots+5n+2 right)2(2n+1)2+ dots=

1+ dots+n+(2n+1)2 left(1+2+3+ dots right)=X.

Arithmetic progression 1+ dots+nis equal to  dfracn(n+1)2, therefore, we obtain the equation


where does it turn out that

X=βˆ’ dfrac18.


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