The limit of sequence is an important concept in mathematics, which describes that a sequence approaches a certain value with infinite terms. In mathematics and application, there are many methods and skills to solve the limit of sequence.
The limit definition method is to solve the limit through the idea that the sequence approaches the limit value item by item. Obey the law of order. The series can be arithmetic progression, geometric progression or other special series. It is necessary to find out the law by observing the first few items of the series for subsequent analysis and calculation. Guess the limit. By observing the law of the sequence, we can preliminarily guess the limit value of the sequence. This guess may be accurate or just an approximation.
Prove it with the definition of limit. According to the definition of limit, for any given positive number ε, there is a positive integer n, so that when n >: when n, the absolute value of the difference between the nth term of the sequence and the limit value is less than ε, that is, |a? -L | & lt； ε. Where, a? Represents the nth term of a series, and l represents the limit value of guess.
Deduction and calculation are carried out according to the guess limit value and limit definition. By deducing the limit definition through deformation, a positive integer n satisfying the conditions can be solved. Use proof conclusion. According to the previous conclusion, we can choose appropriate ε and n, and use the obtained conclusion to prove the limit value of the sequence.
On the solution of sequence of numbers
1, besides the limit definition method, there are other methods to solve the limit of sequence. For example, we can use the properties of sequence, such as recursive relation and mathematical induction, to get the general term formula of sequence, so as to determine the limit of sequence. In addition, the convergence and divergence of series can be used to analyze the limit of series, and the limit of series can be solved by calculating the sum of series.
2. Solving the limit of series is an important content in mathematics, and different series may require different methods and skills. By observing the law of sequence, guessing the limit value and applying the definition of limit to prove it, we can solve the limit of sequence. At the same time, other mathematical tools and concepts can be used to analyze the properties of sequence and further study the limit characteristics of sequence.