Convergence is a process in which the distance between the last term and the previous term of an exponential series is getting smaller and smaller, and finally tends to a fixed value or infinity. In other words, a series or terms of a series are getting closer and closer to a certain value, which is called a limit. For example, the limit 1, 1/2, 1/3, ..., 1/n, ... is 0.
On the contrary, divergence is a process in which the distance between the last term and the previous term of an exponential series is getting bigger and bigger, and it does not tend to any fixed value or infinity. For example, the sequence 1,-1,-1, ..., (-1) n, ... is divergent because it has no fixed boundary.
In mathematical analysis, it is very important to study convergence and divergence, because they can help us understand the behavior of functions and solve some mathematical problems. At the same time, convergence is an important concept in calculus and real number theory.
The main function of convergence in mathematics;
1. Solving Approximate Problems: Many mathematical problems can be solved by finding a series of approximate solutions, and then making them closer to the real solutions. This process usually involves the application of the concept of convergence.
2. Calculating numerical integration: When calculating numerical integration, a method called numerical integration is often used. This method needs to calculate the sum of a series of points to approximate the real integral value, and the summation process of this sum involves the convergence problem.
3. Solving differential equation: In the numerical solution of differential equation, it is often necessary to get the approximate value of the solution through iterative process. The convergence of the iterative process of the solution determines whether the numerical solution is effective or not.
4. Statistical analysis: in statistical analysis, it is often necessary to smooth a group of data to get the centralized trend or average value of the data. This smoothing process involves the problem of convergence. Only when the data converges to a stable value can the effectiveness of smoothing be guaranteed.