1, direct calculation method: for some simple series, the limit value can be directly obtained by calculation. For example, the limit 1, 1/2, 1/3, ... is 0.
2. Pinch theorem: If the sequence {xn} satisfies a≤xn≤b, and the limits of a and b are both L, then the limit of the sequence {xn} is also L, and the pinch theorem can help us to find the limit of the sequence in some cases.
3. Monotonicity is defined as: if the sequence {xn} monotonically increases (or decreases) and has an upper bound (or lower bound), then the sequence {xn} must converge, that is, there is a limit. The monotone bounded theorem can help us prove the convergence of sequence and find the limit value in some cases.
4. Subcolumn method: If a subcolumn {xnk} of the sequence {xn} converges to L, and for any k, there is xn≥xnk (or xn≤xnk), then the sequence {xn} also converges to L. Subcolumn method can help us to find the limit of the sequence in some cases.
5. Stokes Theorem: If the sequence {xn} satisfies xn+ 1-xn≤a(a is a constant), then the sequence {xn} converges. Stokes theorem can help us to prove the convergence of sequence and find the limit value in some cases.
The purpose of this series is:
1, mathematics: series is an important concept in mathematical analysis, which can be used to describe the changing law of a series of values, study the limit, convergence and summation of series, and further deepen the research of mathematical analysis.
2. Physics: Numerical series is also widely used in physics, such as describing the laws of physical quantities such as position, speed and acceleration of an object moving with time, and describing mathematical models of physical phenomena such as fluctuation and vibration.
3. Economy: Data series is often used in economics, for example, to describe the changing law of an indicator with time, such as GDP and price index, and to make predictions and decision analysis.
4. Computer Science: Sequence is also widely used in computer science. For example, in algorithm design and data analysis, it is often necessary to operate and process sequences.