1, limit definition method: limit definition method is the most basic method to judge the convergence of sequence. It judges the convergence of a series by observing that the elements in the series are gradually approaching a certain value. Specifically, for a series {a_n}, if any given positive number ε has a positive integer N, when N is greater than N, the difference between the nth element A _ n in the series and a specific value L is less than ε, then the series is said to converge to L, and it is recorded as lim (a_n) = L.
2. Cauchy convergence criterion: According to this criterion, for any given positive integer ε, there exists a positive integer N. For any natural numbers m and n greater than n, when n and m are large enough, the difference between the nth element a_n and the mth element a_m in the sequence is less than ε. This shows that the elements in the sequence are very close to a limit value after a certain point.
3. Monotone Bounded Principle: Monotone Bounded Principle is also called Monotone Bounded Theorem. The basis for judging the convergence property of sequence is monotonicity and boundedness. The monotone bounded theory points out that if a series monotonically increases and has an upper bound (that is, for any natural number n, a_n ≤ a_{n+ 1} and an upper bound m makes a_n ≤ M), or if a series monotonically decreases and has a lower bound (that is, for any natural number n, a _ n ≥ a _ {n).
The Importance of Learning Advanced Mathematics
Advanced mathematics is a basic subject. By studying advanced mathematics, you will cultivate the ability of logical thinking and problem analysis. Higher mathematics pays attention to the abstraction and reasoning of problems, and cultivates students' abilities of logical thinking, analysis and problem solving, which are essential for the development of various disciplines and occupations.
Advanced mathematics is the foundation of many disciplines, such as physics, engineering, computer science, economics and so on. No matter which field you choose, you need the basic knowledge of advanced mathematics to understand and solve related problems. Calculus, linear algebra and differential equations in advanced mathematics are the core tools and theoretical basis in these disciplines.
Advanced mathematics provides an accurate and quantitative tool for describing and explaining natural and social phenomena. Through the study of advanced mathematics, we can understand and analyze the quantitative relations in physics, chemistry, economy and other fields, and use mathematical models to predict and deduce them. Advanced mathematics helps us understand the laws and complexity of this world.