This is the concrete application of the idea of decomposition and combination in the summation of series. The essence of the split term method is to decompose each term (general term) in the series, and then recombine it, so that it can eliminate some terms and finally achieve the purpose of summation. The general term decomposition (split term) is as follows:

( 1) 1/n(n+ 1)= 1/n- 1/(n+ 1)

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

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

(4) 1/(√a+√b)=[ 1/(a-b)](√a-√b)

(5) n n！ =(n+ 1)！ -No!

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

Solution: Let 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/(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 position of the remaining items is symmetrical before and after.

The positive and negative of the other items are opposite.

Attachment: Common methods for summation of series:

Formula method, split term elimination method, dislocation subtraction method, reverse order addition method, etc. (The key is to find the general item structure of the sequence)

1. Find the sum of series by grouping method: for example, an=2n+3n.

2. Sum by dislocation subtraction: for example, an = n 2n.

3. Sum by split term method: for example, an= 1/n(n+ 1)

4. sum up in reverse order: for example, an = n.

5, the method of finding the maximum and minimum terms of the sequence:

① an+ 1-an = ... For example, an= -2n2+29n-3.

② (An>0) as a =

③ an=f(n) Study the increase and decrease of function f(n), such as an = an 2+bn+c (a ≠ 0).

6. In arithmetic progression, the problem of the maximum value of Sn is usually solved by the adjacent term sign change method;

(1) When A 1 >: 0, d<0, the number m of terms satisfying {an} makes Sm take the maximum value.

(2) when a 1

We should pay attention to the application of the transformation idea when solving the maximum problem of the sequence with absolute value. Quoted from Baidu Encyclopedia