1. Crack propagation

In the summation of series, the summation can be made by the method of split term. Some proof problems involving sequence numbers and inequalities can be summarized by the split term method, and then the sizes of inequalities can be compared.

2. Function scaling

Function scaling is a method to solve the inequality of sequence of numbers by constructing functions and using the monotonicity of functions.

3. Recursive scaling

If the relationship between an and f(n) or an and g(an) is known, we can try to get a geometric series that can be summed by scaling step by step, and then scale the sum result if necessary.

4. Monotonicity scale

For the series inequality starting from n in the form of unilateral summation, we can first construct a monotone series and scale it appropriately by monotonicity, thus proving the inequality skillfully.

5. Strengthen proposition scaling

Because the inequality of sequence is related to positive integers, mathematical induction has become a common method to prove the inequality of sequence, but it is very difficult to prove some inequalities of sequence directly by mathematical induction. At this time, one side of the inequality can be transformed into a geometric series that can be summed, and then it can be proved to be strengthened by mathematical induction.

Local scaling

For many series inequalities whose one side is sum form and the other side is constant, it is usually not necessary to scale from the first term, but to keep the previous term as an accurate value and scale from a certain term, so that the upper (lower) limit estimation on the other side of sum form will be more accurate.