Infinite and finite are essentially different, but they are related. Infinity is limited development. The sum of infinite numbers is not an ordinary algebraic sum. Defining it as the limit of "partial sum" is to know infinity from finite by means of limit method.
"Change" and "invariance" reflect two different states of things, namely, motion change and relative static, but they can be transformed into each other under certain conditions, which is one of the powerful levers of mathematical science. For example, the instantaneous speed of variable-speed linear motion cannot be solved by elementary method, and the difficulty is that the speed is variable at this time. Therefore, people first use uniform speed instead of variable speed in a small range to find its average speed, and define instantaneous speed as average speed.
There are essential differences between curves and straight lines, but they can also be transformed into each other under certain conditions. As Engels said, "Straight lines and curves are finally equal in differential." Making good use of this unity of opposites is one of the important means to deal with mathematical problems. The area of a straight line is easy to get, but we can't just ask for the area of a curve by elementary method. Liu Hui approached a circle inscribed with polygons. Generally people use the area of a small rectangle and the area of a curved trapezoid.
There are differences and connections between quantitative change and qualitative change, and there is a dialectical relationship between them. Quantitative change can lead to qualitative change. The law of mutual change between quality and quantity is one of the basic laws of dialectics and plays an important role in mathematical research. For any circle inscribed with a regular polygon, when the number of its sides is doubled, it will still be inscribed with a regular polygon, which is a quantitative change, not a qualitative change. However, the number of edges will be doubled, and the polygon will change after an infinite process.
Approximation and accuracy are the unity of opposites and can be transformed into each other under certain conditions. This transformation is an important skill of applying mathematics to practical calculation. The aforementioned "partial sum", "average speed" and "the area of a regular polygon inscribed with a circle" are the approximate values of the corresponding infinite series sum, instantaneous speed and circle area in turn, and the corresponding accurate values can be obtained after taking the limit. These are all based on the limit method, and the accuracy can be understood from the approximation.