Important issues and explanations
1, find the limit of sequence.
Finding the limit of sequence can be summarized into the following three forms.
Find the limit of abstract sequence
This kind of questions generally appear in the form of multiple-choice questions, which can be excluded by giving counterexamples. In addition, it can be verified directly according to the definition, basic properties and algorithm.
To find the limit of a specific sequence, you can refer to the following methods:
A, using monotone bounded convergence criterion to find the limit of sequence.
Firstly, the monotonicity and boundedness of sequence are judged by mathematical induction or inequality scale method, and then the existence of limit is judged. Secondly, by taking the limit in the recursive relation and solving the equation, the limit value of the sequence is obtained.
B, using the function limit to find the sequence limit
If sequence limit can be regarded as a special case of function limit, then the relationship between function limit and sequence limit can be transformed into finding function limit, and then it can be solved by Robida's law.
There are several ways to find the limit of the sum or product of terms:
A, using special series summation method
If the general term in the term and limit can be eliminated by dislocation or can be transformed into some form of known limit, then the limit result can be obtained directly by sorting.
B, adopting a power series summation method
If we can find the power series corresponding to this series, we can find its corresponding sum function by the method of power series function, and then substitute this limit form into the corresponding variable to find the function value.
C, using the definition of definite integral to find the limit
If each term of a series can be given a factor and the rest can be expressed by a general term, then the definition of definite integral can be considered to solve the limit of the series.
D, using the pinch theorem to find the limit.
If each term of a series can be given a factor, and the remaining items cannot be expressed by a general term, but the remaining items are arranged in ascending or descending order, we can consider using the pinch theorem to solve it.
E, find the limit of the product of a series of terms, generally in the form of logarithm as the sum of terms, and then calculate by solving the limit of terms and series.