Who can tell me what is the principle of dislocation subtraction and crack elimination of mathematical sequence?

Dislocation subtraction:1(1* 3)+1(3 * 5)+...+1[(2n-1)].

= 1/2( 1- 1/3)+ 1/2( 1/3- 1/5)+...+ 1/[ 1/(2n- 1)- 1/(2n+ 1)]

= 1/2[ 1- 1/(2n+ 1)]

=n/(2n+ 1)

So dislocation subtraction is suitable for the sum of {1[(pn+q) (pn+q+p)]}.

Splitting term elimination method: find the sum of the first n terms of {n * 2 (n- 1)}.

sn= 1* 1+2*2+3*2^2+4*2^3+...+n*2^(n- 1)①

Multiply by 2, and you get

2sn= 1*2+2*2^2+3*2^3+4*2^4+...+(n- 1)2^(n- 1)+n*2^n②

We found that when the right side of Formula ② is moved to the right, compared with Formula ①, the coefficients of 2 ~ 2 (n- 1) are only 1 (see box above), so it can be obtained by subtraction.

-Sn= 1* 1+2+2^2+2^3+...+2^(n- 1)-n*2^n

= 1+2^n-2-n*2^n

=( 1-n)2^n- 1

So sn = (n- 1) 2 n+ 1.

Therefore, the split term elimination method is suitable for the summation of the new series {pnqn} formed by the product of arithmetic progression {pn} and geometric progression {qn}.