1, definition method: for a given series, use definition to judge its convergence and limit value. Definition is one of the most basic methods, which can directly infer the limit from the terms of a series. Discriminant method: use the limit discriminant method to judge the convergence and limit value of the sequence. Criterion methods usually include two main types: Cauchy criterion and Porzano-Weisstras criterion.
2. Property method: Use the property of limit to find the limit of sequence. The properties of limit include: sign preservation, inequality, pinch theorem and so on. Pinch Theorem: When the general term of a sequence can be expressed as the sum or difference of two convergent sequences, we can use the pinch theorem to find its limit.
3. Monotonicity is bounded: if the series monotonically increases (or decreases) and has an upper bound (or lower bound), the series converges and its limit value is equal to the upper bound (or lower bound).
4. Heine theorem: Heine theorem is an important theorem in mathematical analysis, which can be used to solve the limit of sequence. Heine's theorem points out that if each term of a series can be expressed as the sum or difference of infinite arithmetic progression, then the series will converge to a certain value.
5. Compulsive: If there are two series, one of which converges to a certain value and the other converges to another value, the union of these two series converges to the average value of the limit of the first series and the limit of the second series. Series summation method: If a series can be expressed as the sum of the first n terms of an infinite series, the limit of the series can be solved by series summation method.
6. Step-by-step integration method: If a series can be expressed as the sum of the first n terms of a function series, the limit of the series can be solved by step-by-step integration method. L'H?pital's Law: When each term of a series can be expressed as the quotient of two functions, L'H?pital's Law can be used to solve the limit of the series.
The concept of sequence
1, sequence, is an ordered number with positive integer set (or its finite set) as its domain. Every number in a series is called an item of this series, the number ranked first is called 1 item of this series (usually also called the first item), the number ranked second is called the second item of this series, and so on.
2. At the same time, the sequence can also be regarded as a function defined on the set of discrete numbers, which is the basis of discrete mathematics and the basis of defining limits. The relationship between sequence and function is very close, and properties such as monotonicity and maximum can be studied from the perspective of function.