Split term formula for summation of sequence

Split term formula for summation of sequence

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Sum of split term method
This is the concrete application of the idea of decomposition and combination in the summation of sequence. The essence of split term method is to decompose each term (general term) in the sequence, and then recombine them, so that some terms can be eliminated, and finally the purpose of summation can be achieved
　　（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)!-n!
[example 1] [basic type of fractional split term] finding the sum of the first n terms of a sequence an = 1 / N (n + 1)
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) (sum of split terms)
　　＝ 1-1/(n+1)
　　＝ n/(n+1)
[example 2] [basic form of integer split term] find the sum of the first n terms of a sequence an = n (n + 1)
An = n (n + 1) = [n (n + 1) (n + 2) - (n-1) n (n + 1)] / 3 (split term)
Then Sn = [1 × 2 × 3-0 × 1 × 2 + 2 × 3 × 4-1 × 2 × 3 + +N (n + 1) (n + 2) - (n-1) n (n + 1)] / 3 (sum of split terms)
　　＝ (n-1)n(n+1)/3
Summary: the characteristic of this kind of deformation is that after splitting each item of the original sequence into two items, most of the items in the middle cancel each other, leaving only a few items
Note: the remaining items have the following characteristics
The positions of the remaining items are symmetrical
The positive and negative of the remaining items are opposite
Error prone point: pay attention to check whether the formula after the split term is equal to the original formula. Typical errors are: 1 / (3 × 5) = 1 / 3-1 / 5 (the right side of the equation should be divided by 2)
Appendix: common methods of sequence summation:
The key is to find the general term structure of sequence
1. Group method to find the sum of sequence: for example, an = 2n + 3N
2. Sum by dislocation subtraction method: for example, an = n · 2 ^ n
3. Sum by split term method: for example, an = 1 / N (n + 1)
4. Sum of reverse order addition: 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
(an > 0) such as an=
(3) an = f (n) study the increase and decrease of function f (n), such as an = an ^ 2 + BN + C (a ≠ 0)
6. In the arithmetic sequence, the problem of the maximum value of Sn is often solved by the adjacent term sign changing method
(1) when A1 > 0, D