Example:
Total:1/2+1/6+112+1/20.
= 1/( 1*2)+ 1/(2*3)+ 1/(3*4)+ 1/(4*5)
=( 1- 1/2)+( 1/2- 1/3)+( 1/3- 1/4)+( 1/4- 1/5)
= 1- 1/2+ 1/2- 1/3+ 1/3- 1/4+ 1/4- 1/5
= 1- 1/5=4/5
It is known that the sum of ampere (series) is the sum of the most common split terms, which is a kind of regular problem and exists universally in senior two mathematics.
First, the basic concept:
1, definition and representation of sequence:
2. Items and number of items in the series:
3, finite sequence and infinite sequence:
4, increasing (decreasing), swing, cycle order:
5. The general formula of sequence {an} an:
6. The first n terms of the sequence and the formula Sn:
7. Structure of arithmetic progression, Tolerance D and arithmetic progression:
8. The structure of geometric series, Bi Gong Q and geometric series;
Second, the basic formula:
9. the relationship between the general term an and the first n terms and Sn of a general sequence: an=
10, the general formula of arithmetic progression: an = a 1+(n-1) Dan = AK+(n-k) d (where a1is the first term and AK is the known k term), when d≠0.
1 1, the first n terms of arithmetic progression and its formula: Sn= Sn= Sn=
When d≠0, Sn is a quadratic form about n, and the constant term is 0; When d=0 (a 1≠0), Sn=na 1 is a proportional formula about n. 。
12, the general formula of geometric series: an = a1qn-1an = akqn-k.
(where a 1 is the first term, ak is the known k term, and an≠0).
13, the first n terms of geometric series and their formulas: when q= 1, Sn=n a 1 (this is a direct ratio formula about n);
When q≠ 1, Sn= Sn=
Third, the conclusion about arithmetic and geometric series.
Arithmetic progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 14 is still arithmetic progression.
15, arithmetic progression {an}, if m+n=p+q, then
16, geometric series {an}, if m+n=p+q, then
Geometric progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 17 is still geometric progression.
18, the sum and difference of two arithmetic progression {an} and {bn} series {an+bn} is still arithmetic progression.
19, a sequence consisting of the product, quotient and reciprocal of two geometric series {an} and {bn}
{an bn},, or geometric series.
20. arithmetic progression {an} Any equidistant series is still arithmetic progression.
2 1, the series of any equidistant term of geometric progression {an} is still geometric progression.
22. How to make three numbers equal: A-D, A, A+D; The method of four numbers being equal: a-3d, a-d, a+d, a+3d.
23. How to make three numbers equal: A/Q, A, AQ;
Wrong method of four numbers being equal: a/q3, a/q, aq, aq3 (Why? )
24.{an} is arithmetic progression, then (c>0) is a geometric series.
25 、{ bn }(bn & gt； 0) is a geometric series, then {logcbn} (c >; 0 and c 1) are arithmetic progression.
26. In the arithmetic series:
(1) If the number of items is, then
(2) If the quantity is,
27. In geometric series:
(1) If the number of items is, then
(2) If the number is 0,
Four, the common methods of sequence summation: formula method, split item elimination method, dislocation subtraction, anti-addition, etc. The key is to find the general term structure of series.
28. Find the sum of series by grouping method: for example, an=2n+3n.
29. Sum by dislocation subtraction: for example, an=(2n- 1)2n.
30. Sum by split term method: for example, an= 1/n(n+ 1).
3 1, sum by addition in reverse order: for example, an=
32. The method of finding the maximum and minimum term of series {an}:
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=
33. In arithmetic progression, the problem of the maximum value of Sn is often solved by the adjacent term sign change method.
(1) when >: 0, d