1. Direct method: For a given sequence, we can directly judge whether it converges by observing the values of some items. If some terms tend to a fixed value, the sequence is convergent. This method is suitable for simple series, such as arithmetic progression and geometric progression.
2. Limit method: For a given series, we can judge whether it converges by finding the limit of some items. If the limits of some items in a series exist and are equal to the same value, then the series is convergent. This method is suitable for general sequences, but it should be noted that when the limits of some items in the sequence do not exist, it is impossible to directly judge whether the sequence converges.
3. Pinch theorem: Pinch theorem is a common method to solve the convergence domain of sequence. Its basic idea is that if a sequence is squeezed by two known convergent sequences, then this sequence must also converge. Through the pinch theorem, the calculation process of sequence convergence region can be simplified.
4. Monotone Bounded Criterion: Monotone Bounded Criterion is a common method to solve the convergence domain of sequences. Its basic idea is that if a sequence is monotonically bounded, then it must converge. The monotone bounded criterion can be used to quickly judge the convergence of the sequence.
5. Using the known convergence sequence: When solving the convergence domain of the sequence, the known convergence sequence can be used to assist the judgment. For example, if a sequence can be expressed as the sum or difference of known convergent sequences, then this sequence must also converge.
6. Use mathematical software: For complex sequences, mathematical software (such as MATLAB, Mathematica, etc. ) can be used to solve its convergence domain. These softwares usually have a wealth of built-in mathematical functions and algorithms, which can easily solve various mathematical problems.