Two common methods to prove monotonicity of sequence are xn+1-xn > =0 or = 1.

I. Analysis

Xn+ 1-Xn & gt； 0 or = 1 or xn/xn+1>; = 1 and sequence monotonicity are necessary and sufficient conditions.

1. For the sequence itself, monotonically increasing sequence is defined as all items are positive numbers, and each item from left to right is greater than the previous item; While monotonically decreasing, each term is negative. From left to right, each item is smaller than the previous one.

2. The ratio of the previous term to the latter term of a sequence can always be used to judge whether the sequence is monotonous. Monotonically increasing sequence, each term of which is greater than the previous term; When monotonically decreasing, each term is smaller than the previous one.

Second, the concept of sequence

Literally, a series is a series of numbers arranged in succession, which is the literal meaning. The mathematical definition is that a series of numbers arranged in a certain order is called a sequence.

Third, the classification of series.

1, according to the series of series items, the number of series items is limited, such as ten items, one hundred items and ten thousand items, which can be countable or infinite.

2. According to the changing trend of the series project size, it can be divided into increasing series, decreasing series, constant series and swinging series (the changing trend of the project size is not fixed, maybe this project is larger than the latter one or smaller than the latter one).

series formula

1, arithmetic progression

Arithmetic progression is a series with equal difference between two adjacent terms in an exponential series, and its general term formula is:

an = a 1+(n- 1)d = Sn-S(n- 1)(n≥2)= kn+b .

sn = n(a 1+an)/2 = na 1+n(n- 1)d/2 .

an=am+(n-m)d .

2. Equal ratio series

Geometric series is a series with equal ratio between two adjacent terms in an exponential series, and its general term formula is:

an=a 1q^(n- 1)=sn-s(n- 1)(n≥2)。

Sn = a1(1-q n)/(1-q) = (a1-anq)/(1-q) (q ≠1).

an=amq^(n-m)。

3. Fibonacci series formula

Fibonacci series refers to a series in which each term in the exponential series is equal to the sum of the first two terms, and its general term formula is:

An =an- 1+an-2, where A 1 = 1, A2 = 1, n >；; =3。

Fibonacci sequence also has the golden section formula P =( 1+V5)/2.

Where p is the golden ratio.

4. Harmonic sequence formula

A harmonic sequence refers to a sequence in which the sum of the reciprocal of each term in an exponential sequence is equal to a constant, and its general term formula is:

An = 1/n, where n is the number of terms.

Harmonic sequence and summation formula:

sn = 1/ 1+ 1/2+ 1/3+…？ + 1/n .

Where Sn is the sum of the first n terms.