General term formula applicable to fractional form. A term is divided into two or more difference forms, that is, an = f (n+ 1)-f (n), and then many intermediate terms are cancelled when accumulating.

Commonly used formula:

( 1)

(2)

(3)

(4) (when a≠b)

(5)

[Example] Find the sum of the first n items in a series an= 1/n(n+ 1).

Solution: an =1/n (n+1) =1/n-1/(n+1) (split term)

Then Sn

= 1- 1/2+ 1/2- 1/3+ 1/4 ...+ 1/n- 1/(n+ 1)

= 1- 1/(n+ 1)

= n/(n+ 1)

Summary: This deformation is characterized by the fact that after each item in the original series is split into two items, most of the items in the middle cancel each other out. There are only a few things left.

Note: The remaining projects have the following characteristics.

1. The positions before and after the other items are symmetrical.

2. The positivity and negativity of other projects are opposite.