1. Limit definition:
According to the limit definition of function, we can judge the convergence and divergence of function by finding the limit value of function at a certain point or an interval. If the limit of a function exists and is finite at this point or interval, the function is convergent. If the limit of the function at this point or interval does not exist or tends to infinity, the function diverges.
2. Convergence criteria of sequences;
For real functions, the convergence and divergence of functions can be judged by the convergence criterion of sequence. This method transforms a function into a series of numbers for judgment. If there is a convergent sequence, the function converges when the limit of the sequence is equal to the limit of the function. If all series have no convergence limit, or at least one series exists, so that its limit is infinite or nonexistent, then this function is divergent.
3. Monotonicity and boundedness;
If a function monotonically increases or decreases in an interval, and there is an upper bound or a lower bound in this interval, then the function is convergent. A function is divergent if it is neither monotonous nor bounded in a certain interval.
4. Derivative and differential:
For differentiable functions, the convergence and divergence of functions can be judged by the properties of derivatives. If the derivative of a function exists and is finite at a certain point, then the function converges near that point. If the derivative of a function tends to infinity or does not exist, the function diverges.
5. Behavior at infinity:
By studying the limit behavior of a function at infinity, we can judge the convergence and divergence of the function. For example, a function is convergent if its limit at infinity exists and is finite. A function is divergent if its limit at infinity is infinite or nonexistent.
Expand knowledge:
The concept of convergence and divergence is a basic concept in mathematical analysis, which is widely used in calculus, series, sequence and other fields. The convergence of functions is of great significance in practical problems, for example, it is often necessary to judge the convergence of functions in numerical calculation and mathematical modeling.
The methods of judging the convergence and divergence of functions are not independent of each other, and many methods can be used to judge the properties of functions. The convergence and divergence of functions is the basic concept of mathematical analysis. Understanding and mastering these concepts in depth is helpful to understand and apply more advanced mathematical concepts and methods.