This is the concrete application 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 sequence, and then recombine it, so that some terms can be eliminated and finally the purpose of summation can be achieved.
General term decomposition (split term), such as:
( 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)
No, no! =(n+ 1)！ -No!
[Example 1]
Find the sequence an= 1/n(n+ 1)
The sum of the first n items of.
Solution: an =1/n (n+1) =1/n-1(n+1)
(crack)
rule
sn = 1- 1/2+ 1/2- 1/3+ 1/4...+ 1/n- 1/(n+ 1
=
1- 1/(n+ 1)
=
n/(n+ 1)
[Example 2]
According to the basic form of integer splitting term, find the sequence an=n(n+ 1).
The sum of the first n items of.
Solution: an = n (n+1) = [n (n+1) (n+2)-(n-1) n (n+1)]/3 (split term)
rule
sn =[ 1×2×3-0× 1×2+2×3×4- 1×2×3+...+N (n+ 1) (n+2)-(n-65438)
=
(n- 1)n(n+ 1)/3
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.
Error-prone point: Pay attention to check whether the formula after the split item is equal to the original formula. The typical error is:1/(3× 5) =1/3-1/5 (the right side of the equation should be divided by 2).
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. Add and sum 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, a =
-2n2+29n-3
②
(An>0)
For example, a =
③
an=f(n)
Study the increase and decrease of function f(n)
For example, a =
an^2+bn+c(a≠0)
6. In arithmetic series
About Sn
-the maximum value problem solved by the commonly used adjacent term sign change method;
(1) When
a 1 & gt； 0, d<0, the number m of terms satisfying {an} makes Sm take the maximum value.
(2) When
a 1 & lt； 0, d>0, the number m of terms satisfying {an} makes Sm the smallest.
We should pay attention to the application of the transformation idea when solving the maximum problem of the sequence with absolute value.