In mathematics, convergence and divergence are properties of exponential series or series. Judging whether a sequence or series converges or diverges is an important problem in mathematics learning. Here are some formulas to judge the convergence and divergence of a sequence or series.

First, the convergence formula of sequence.

1, monotone bounded principle: If a sequence monotonically increases and has an upper bound, or monotonically decreases and has a lower bound, then the sequence must converge.

2. Pinch criterion: If a series is between two convergent series, then this series also converges.

3. Limit algorithm: If the limit of a series exists, then the series must converge.

Second, the divergence formula of series.

1, the general term tends to infinity: if the general term of a series tends to positive infinity or negative infinity, then the series diverges.

2. Oscillation divergence: If a series oscillates back and forth between two numbers, then the series is divergent.

3. Infinite approximation: If the general term of a series approaches a certain number infinitely, but it is not equal to this number, then the series diverges.

Third, the formula of series convergence.

1, comparison method: If the general term of one series can be compared with the general term of another series and this series converges, then this series also converges.

2. Ratio discrimination method: If the ratio of the absolute value of the general term of a series tends to 0, the series converges.

3. Root value discrimination method: If the root value of the absolute value of the general term of a series tends to 0, the series converges.

Fourth, the formula of series divergence.

1, positive series: If all the general terms of a series are positive and the series diverges, then the series must tend to positive infinity.

2. Comparative judgment method: If the general term of one series can be compared with the general term of another series, and this series diverges, then this series also diverges.

3. Ratio discrimination method: If the ratio of the absolute value of the general term of a series tends to positive infinity or negative infinity, the series diverges.

The benefits of mathematics

1. Mathematics is the foundation of all sciences, and all major scientific and technological progress is closely related to mathematics. Without mathematics, there would be no computers, televisions and space shuttles, and there would be no such colorful life today.

2. Mathematics is a tool discipline, which is the basis of learning other disciplines and also a discipline to improve people's judgment, analysis and understanding.

Mathematics is not only a science, but also a universally applicable technology. It is the door and key to science. Learning mathematics is a very important measure to make oneself rational, and mathematics itself has its fun.

4, mathematics can make thinking about any problem more detailed, and there will be no thinking obstacles. It can also make the brain reflect flexibly and deal with emergencies more rationally.

5. Mathematics gives people not only knowledge, but also ability, including observing experiments, collecting information, inductive analogy, intuitive judgment, logical reasoning, modeling and accurate calculation. These abilities and training will benefit people for life.