Hans Walser, [20260617]

Sequence and Limit

1 The Sequence

We work with the sequence:

(1)

Table 1 shows the first 30 elements of the sequence:

n	an	an	n	an	an	n	an	an
1	13/3	4.333333333	11	43/13	3.307692308	21	73/23	3.173913043
2	4	4.	12	23/7	3.285714286	22	19/6	3.166666667
3	19/5	3.800000000	13	49/15	3.266666667	23	79/25	3.160000000
4	11/3	3.666666667	14	13/4	3.250000000	24	41/13	3.153846154
5	25/7	3.571428571	15	55/17	3.235294118	25	85/27	3.148148148
6	7/2	3.500000000	16	29/9	3.222222222	26	22/7	3.142857143
7	31/9	3.444444444	17	61/19	3.210526316	27	91/29	3.137931034
8	17/5	3.400000000	18	16/5	3.200000000	28	47/15	3.133333333
9	37/11	3.363636364	19	67/21	3.190476190	29	97/31	3.129032258
10	10/3	3.333333333	20	35/11	3.181818182	30	25/8	3.125000000

Table 1: First terms of the sequence

Figure 1 shows a corresponding point representation.

Fig. 1: Point Representation

2 Transformation

We transform:

(2)

The trick is reminiscent of "completing the square" when solving a quadratic equation.

Hence:

(3)

3 Limit and Monotony

The second term of (3) is a monotonically decreasing null sequence. Therefore, the constant first term 3 is the limit of the sequence. The sequence is monotonically decreasing.