There is a sequence

, , , ,

Find .

　　Solution: Assume that the sequence is a recursive sequence. The existence of the limit of such a sequence is often judged by the monotonic bounded criterion. For this question, obviously, there are

,

Therefore, the sequence increases monotonically.

　　The following uses mathematical induction to prove that the sequence is bounded:

　　Because , assuming , then we have

.

Therefore is bounded. According to criterion II, exists.

　　Suppose , we know from the sign preservation property of the limit of the sequence, , because

,

Right now

,

Therefore, taking the limits on both ends of the above equation , we get

,

Right now

,

Solving the above quadratic equation, we get

or ( reject ) .

so

.

　　For the graphics experiment of this example, see Experiment 1.9 of §1.3 .