"convergent sequences is a mathematical term, set the sequence Xn, if there is a constant a (only one), for any given positive number q (no matter how small), there is always a positive integer n, which makes N >；; When n, there is always | Xn-a | < Q, which means that the sequence xn converges to a (the limit is a), that is, the number xn is a convergent sequence.

Convergence is a general term for sequences. When n tends to infinity, the general term of the sequence tends to a number, that is, there is a limit. In fact, high school mathematics is very simple, and only simple subtraction and increase are learned in the series.

The convergence of series has nothing to do with the previous finite term: that is, removing the finite term or adding the finite term will not affect the convergence of series; If the sequence converges, it does not affect the limit value of the sequence.

Boundedness of convergent sequence: if the sequence {an} converges to a, then the sequence {an} is bounded, that is, there is m >；; 0, so |an|≤M holds.

The relationship between convergent sequence and its subsequence;

A subsequence is also a convergent sequence, and the limit is constant | xn | < m. If one subsequence is known to be divergent or two subsequences converge to different limit values, it can be concluded that the original sequence is divergent. If a sequence converges to a, then any of its subsequences also converges to a.

This also shows that:

1. If the sequence {an} converges to a, for any given positive number ε, at most only the finite term of an falls outside the neighborhood U(a, ε) with A as the center and ε as the radius.

2. If the series {an} converges to a, there must be a maximum number or a minimum number in this series, but not necessarily both.

3. The convergence of series must be bounded, but the bounded series may not necessarily converge! Number-preserving property of convergent sequence: (1) If an≥0 and the sequence {an} converges to a, then a≥0.

Interrelation:

1, the relationship between convergent sequence and its subsequence

2. The subsequence is also a convergent sequence, and the limit is constant | xn | < m.

3. If one subsequence is known to be divergent, or two subsequences converge to different limit values, it can be inferred that the original sequence is divergent.

4. If the sequence {} converges to A, then any of its subsequences also converges to A. ..