The judgment theorem of convergence and divergence of high number series is an important tool for studying infinite series in higher mathematics, which is used to judge whether a given infinite series converges or not. The following are some common theorems for judging the convergence and divergence of high-order series:

1. Comparative judgment method: If an infinite series has the same form as another series with known convergence and divergence, and their terms can be compared item by item, then the convergence and divergence of this series is the same as that of the known series.

2. Ratio discrimination method: If the ratio of two adjacent terms of an infinite series tends to be a constant, then the series may converge or diverge. If this constant is less than 1, then the series converges; If this constant is greater than 1, then the series diverges; If this constant is equal to 1, then it needs to be further judged by other methods.

3. Root value discrimination method: If every term of an infinite series is positive and its reciprocal forms a monotonically decreasing sequence, then the series may be convergent or divergent. If the maximum value of this series is less than 1, then this series converges; If the maximum value is greater than 1, the series diverges; If the maximum value is equal to 1, other methods need to be further used for judgment.

4. Integral discriminant method: For positive series, we can use integral discriminant method to judge its convergence and divergence. Specifically, each term of the series is integrated after taking an absolute value. If the integration converges, the original series also converges. If the integral diverges, the original series also diverges.

5. Pinch Theorem: If an infinite series is sandwiched by two known convergent or divergent series, that is, sandwiched between these two series, then the convergence and divergence of this series are the same as those of the known series.

6. Limit discrimination method: For positive series, the convergence and divergence can be judged by calculating the limit of partial sum. If the limit of partial sum exists and is finite, the series converges; If the limit of partial sum does not exist or is infinite, the series diverges.

The above are the common judgment theorems of convergence and divergence of high number series, which are very useful in solving practical problems. By applying these theorems, we can judge whether a given infinite series converges, so as to make further analysis and calculation.