Convergence means that 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.

This also shows that:

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

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

(3)? The convergence of the sequence must be bounded, but the bounded sequence does not necessarily converge!

Number-preserving property of convergent sequence: (1) If an≥0 and the sequence {an} converges to a, then a≥0.