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

Many people know that



But in reality



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



using the cesaro method will be equal $\ dfrac12$.

Another example:



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 $- \ dfrac {1} {12}$) 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$$X$:



Now we perform the following transformations:





From here



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





i.e



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



$1+2+3+49X=X,$

which will give the same result.

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





Arithmetic progression $1 + \ dots + n$is equal to $\ dfrac {n (n + 1)} {2}$, therefore, we obtain the equation



where does it turn out that



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