Computable sums are convergent and irreconcilable sums are divergent.

The essence of the problem is to prove that the sequence Sn =1+1/2+1/3+...+1/n is divergent.

Proof process

Take n arbitrarily so that m=2n, and use.

{sm-sn} =1/(n+1)+1(n+2)+...+1/(n+n) greater than or equal to1/(n+n).

, let a= 1/2, for any n, when n >; stone

The absolute value of S2n-Sn is greater than a= 1/2.

Cauchy convergence criterion shows that Sn = {1+1/2+1/3+...+1/n} diverges.

enclose herewith

Cauchy convergence criterion

The necessary and sufficient condition for the convergence of sequence is

For any number greater than 0

There is a number n greater than 0, so

M, n> Sometimes.

The absolute value of Sn-Sm is less than a.

This standard is understandable.

The values of the items in the convergence sequence get closer and closer, so the absolute value of the difference between them can be less than any given positive number.

That is, Sn is a divergent sequence.

No boundaries are irreconcilable.

S 1+S2+S3+……+S(n-2)=[S(n- 1)- 1]g(n)

S 1+S2+S3+……+S(n- 1)=(Sn- 1)g(n)

You can get [sn-s (n-1)] g (n) = s (n-1).