Theory and Modern Applications
Table 6 Some numerical results of the \(\Delta ^{(i)}\) and K in Example 1 for \(q=\frac{1}{7}, \frac{1}{2}, \frac{7}{8}\)
n | \(\Delta ^{(1)}\) | \(\Delta ^{(2)}\) | \(\Delta ^{(3)}\) | K | |
|---|---|---|---|---|---|
\(q =\frac{1}{7}\) | |||||
1 | 6.8851 | 7.1054 | 1.7625 | 0.9846 | |
2 | 6.8883 | 7.1084 | 1.7611 | 0.9849 | |
3 | 6.8888 | 7.1089 | 1.7609 | 0.9849 | |
4 | 6.8888 | 7.1089 | 1.7608 | 0.9849 | |
5 | 6.8888 | 7.1089 | 1.7608 | 0.9849 | |
6 | 6.8888 | 7.1089 | 1.7608 | 0.9849 | |
\(q =\frac{1}{2}\) | |||||
1 | 4.9587 | 5.4326 | 1.4216 | 0.7383 | |
2 | 5.0932 | 5.5666 | 1.4203 | 0.755 | |
3 | 5.1594 | 5.6326 | 1.4196 | 0.7632 | |
4 | 5.1922 | 5.6653 | 1.4192 | 0.7673 | |
5 | 5.2086 | 5.6816 | 1.419 | 0.7693 | |
6 | 5.2168 | 5.6898 | 1.4189 | 0.7703 | |
7 | 5.2209 | 5.6938 | 1.4188 | 0.7708 | |
8 | 5.2229 | 5.6959 | 1.4188 | 0.7711 | |
9 | 5.2239 | 5.6969 | 1.4188 | 0.7712 | |
10 | 5.2245 | 5.6974 | 1.4188 | 0.7713 | |
11 | 5.2247 | 5.6976 | 1.4188 | 0.7713 | |
12 | 5.2248 | 5.6978 | 1.4188 | 0.7713 | |
13 | 5.2249 | 5.6978 | 1.4188 | 0.7713 | |
14 | 5.2249 | 5.6979 | 1.4188 | 0.7713 | |
15 | 5.2249 | 5.6979 | 1.4188 | 0.7714 | |
16 | 5.225 | 5.6979 | 1.4188 | 0.7714 | |
17 | 5.225 | 5.6979 | 1.4188 | 0.7714 | |
18 | 5.225 | 5.6979 | 1.4188 | 0.7714 | |
\(q =\frac{7}{8}\) | |||||
1 | 2.4252 | 2.8232 | 0.8529 | 0.3813 | |
2 | 2.701 | 3.1283 | 0.9157 | 0.4216 | |
3 | 2.9255 | 3.3747 | 0.9626 | 0.4539 | |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
64 | 4.3002 | 4.8561 | 1.1912 | 0.6467 | |
65 | 4.3002 | 4.8562 | 1.1912 | 0.6467 | |
66 | 4.3003 | 4.8562 | 1.1913 | 0.6467 | |
67 | 4.3003 | 4.8562 | 1.1913 | 0.6467 | |
68 | 4.3003 | 4.8563 | 1.1913 | 0.6467 | |
69 | 4.3004 | 4.8563 | 1.1913 | 0.6467 | |
70 | 4.3004 | 4.8563 | 1.1913 | 0.6467 | |
71 | 4.3004 | 4.8563 | 1.1913 | 0.6468 | |
72 | 4.3004 | 4.8564 | 1.1913 | 0.6468 | |
73 | 4.3005 | 4.8564 | 1.1913 | 0.6468 | |
74 | 4.3005 | 4.8564 | 1.1913 | 0.6468 | |
75 | 4.3005 | 4.8564 | 1.1913 | 0.6468 | |