问通项公式

王守恩

王守恩 发表于 2021-1-12 08:47
谢谢 elim！
\(在这里，我们可以把\ \infty\ 去掉。\)
\(下面的题，我们还可以把\ \infty\ 去掉吗？\ ...

\(\ 10楼的\ \infty\ 不能去掉。但下面的\ \infty\ 可以去掉，且\ 2\ 可改任意数。\)
\(\displaystyle1，\sum_{k=2}^{\infty}\frac{1}{k}-\sum_{k=A}^{\infty/A}\frac{1}{k}\equiv\sum_{k=1}^{A-2}\ \frac{k}{(A-1)(A-1-k)}+\ln A\)
\(\displaystyle2，\sum_{k=2}^{\infty/A}\frac{1}{k}-\sum_{k=A}^{\infty}\frac{1}{k}\equiv\sum_{k=1}^{A-2}\ \frac{k}{(A-1)(A-1-k)}-\ln A\)

王守恩

elim 发表于 2021-1-2 11:14
\(\displaystyle\lim_{N\to\infty}\prod_{k=1}^N\frac{a+\lambda^{k+d}}{a+\lambda^{k}}=\lim_{N\to\infty} ...

{2, 12, 212, 1212, 21212, 121212, 2121212, 12121212, 212121212, 1212121212, 21212121212, 121212121212, 2121212121212, 12121212121212, 212121212121212, 1212121212121212},
{3, 23, 323, 2323, 32323, 232323, 3232323, 23232323, 323232323, 2323232323, 32323232323, 232323232323, 3232323232323, 23232323232323, 323232323232323, 2323232323232323},
{4, 34, 434, 3434, 43434, 343434, 4343434, 34343434, 434343434, 3434343434, 43434343434, 343434343434, 4343434343434, 34343434343434, 434343434343434, 3434343434343434},
{5, 45, 545, 4545, 54545, 454545, 5454545, 45454545, 545454545, 4545454545, 54545454545, 454545454545, 5454545454545, 45454545454545, 545454545454545, 4545454545454545},
{6, 56, 656, 5656, 65656, 565656, 6565656, 56565656, 656565656, 5656565656, 65656565656, 565656565656, 6565656565656, 56565656565656, 656565656565656, 5656565656565656},
{7, 67, 767, 6767, 76767, 676767, 7676767, 67676767, 767676767, 6767676767, 76767676767, 676767676767, 7676767676767, 67676767676767, 767676767676767, 6767676767676767},
{8, 78, 878, 7878, 87878, 787878, 8787878, 78787878, 878787878, 7878787878, 87878787878, 787878787878, 8787878787878, 78787878787878, 878787878787878, 7878787878787878},
{9, 89, 989, 8989, 98989, 898989, 9898989, 89898989, 989898989, 8989898989, 98989898989, 898989898989, 9898989898989, 89898989898989, 989898989898989, 8989898989898989}}

Table[((11 a + 1) (10^n - 1) + 9 Mod[n, 2] 10^n)/99, {a, 8}, {n, 16}]

王守恩

{2, 12, 212, 1212, 21212, 121212, 2121212, 12121212, 212121212, 1212121212, 21212121212, 121212121212, 2121212121212, 12121212121212, 212121212121212, 1212121212121212},
{3, 23, 323, 2323, 32323, 232323, 3232323, 23232323, 323232323, 2323232323, 32323232323, 232323232323, 3232323232323, 23232323232323, 323232323232323, 2323232323232323},
{4, 34, 434, 3434, 43434, 343434, 4343434, 34343434, 434343434, 3434343434, 43434343434, 343434343434, 4343434343434, 34343434343434, 434343434343434, 3434343434343434},
{5, 45, 545, 4545, 54545, 454545, 5454545, 45454545, 545454545, 4545454545, 54545454545, 454545454545, 5454545454545, 45454545454545, 545454545454545, 4545454545454545},
{6, 56, 656, 5656, 65656, 565656, 6565656, 56565656, 656565656, 5656565656, 65656565656, 565656565656, 6565656565656, 56565656565656, 656565656565656, 5656565656565656},
{7, 67, 767, 6767, 76767, 676767, 7676767, 67676767, 767676767, 6767676767, 76767676767, 676767676767, 7676767676767, 67676767676767, 767676767676767, 6767676767676767},
{8, 78, 878, 7878, 87878, 787878, 8787878, 78787878, 878787878, 7878787878, 87878787878, 787878787878, 8787878787878, 78787878787878, 878787878787878, 7878787878787878},
{9, 89, 989, 8989, 98989, 898989, 9898989, 89898989, 989898989, 8989898989, 98989898989, 898989898989, 9898989898989, 89898989898989, 989898989898989, 8989898989898989}}

Table[((11 a + 1) (10^n - 1) + 9 Mod[n, 2] 10^n)/99, {a, 8}, {n, 16}]

{11, 1011, 111011, 10111011, 1110111011, 101110111011, 11101110111011, 1011101110111011, 111011101110111011, 10111011101110111011, 1110111011101110111011, 101110111011101110111011},
{12, 1112, 121112, 11121112, 1211121112, 111211121112, 12111211121112, 1112111211121112, 121112111211121112, 11121112111211121112, 1211121112111211121112, 111211121112111211121112},
{13, 1213, 131213, 12131213, 1312131213, 121312131213, 13121312131213, 1213121312131213, 131213121312131213, 12131213121312131213, 1312131213121312131213, 121312131213121312131213},
{14, 1314, 141314, 13141314, 1413141314, 131413141314, 14131413141314, 1314131413141314, 141314131413141314, 13141314131413141314, 1413141314131413141314, 131413141314131413141314},
{15, 1415, 151415, 14151415, 1514151415, 141514151415, 15141514151415, 1415141514151415, 151415141514151415, 14151415141514151415, 1514151415141514151415, 141514151415141514151415},
{16, 1516, 161516, 15161516, 1615161516, 151615161516, 16151615161516, 1516151615161516, 161516151615161516, 15161516151615161516, 1615161516151615161516, 151615161516151615161516},
{17, 1617, 171617, 16171617, 1716171617, 161716171617, 17161716171617, 1617161716171617, 171617161716171617, 16171617161716171617, 1716171617161716171617, 161716171617161716171617},
{18, 1718, 181718, 17181718, 1817181718, 171817181718, 18171817181718, 1718171817181718, 181718171817181718, 17181718171817181718, 1817181718171817181718, 171817181718171817181718}}

Table[((101 (a + 9) + 1) (100^n - 1) + 99 Mod[n, 2] 100^n)/9999, {a, 8}, {n, 12}]

{101, 100101, 101100101, 100101100101, 101100101100101, 100101100101100101, 101100101100101100101, 100101100101100101100101, 101100101100101100101100101, 100101100101100101100101100101},
{102, 101102, 102101102, 101102101102, 102101102101102, 101102101102101102, 102101102101102101102, 101102101102101102101102, 102101102101102101102101102, 101102101102101102101102101102},
{103, 102103, 103102103, 102103102103, 103102103102103, 102103102103102103, 103102103102103102103, 102103102103102103102103, 103102103102103102103102103, 102103102103102103102103102103},
{104, 103104, 104103104, 103104103104, 104103104103104, 103104103104103104, 104103104103104103104, 103104103104103104103104, 104103104103104103104103104, 103104103104103104103104103104},
{105, 104105, 105104105, 104105104105, 105104105104105, 104105104105104105, 105104105104105104105, 104105104105104105104105, 105104105104105104105104105, 104105104105104105104105104105},
{106, 105106, 106105106, 105106105106, 106105106105106, 105106105106105106, 106105106105106105106, 105106105106105106105106, 106105106105106105106105106, 105106105106105106105106105106},
{107, 106107, 107106107, 106107106107, 107106107106107, 106107106107106107, 107106107106107106107, 106107106107106107106107, 107106107106107106107106107, 106107106107106107106107106107},
{108, 107108, 108107108, 107108107108, 108107108107108, 107108107108107108, 108107108107108107108, 107108107108107108107108, 108107108107108107108107108, 107108107108107108107108107108}}

Table[((1001 (a + 99) + 1) (1000^n - 1) + 999 Mod[n, 2] 1000^n)/999999, {a, 8}, {n, 10}]

{1001, 10001001, 100110001001, 1000100110001001, 10011000100110001001, 100010011000100110001001, 1001100010011000100110001001, 10001001100010011000100110001001, 100110001001100010011000100110001001},
{1002, 10011002, 100210011002, 1001100210011002, 10021001100210011002, 100110021001100210011002, 1002100110021001100210011002, 10011002100110021001100210011002, 100210011002100110021001100210011002},
{1003, 10021003, 100310021003, 1002100310021003, 10031002100310021003, 100210031002100310021003, 1003100210031002100310021003, 10021003100210031002100310021003, 100310021003100210031002100310021003},
{1004, 10031004, 100410031004, 1003100410031004, 10041003100410031004, 100310041003100410031004, 1004100310041003100410031004, 10031004100310041003100410031004, 100410031004100310041003100410031004},
{1005, 10041005, 100510041005, 1004100510041005, 10051004100510041005, 100410051004100510041005, 1005100410051004100510041005, 10041005100410051004100510041005, 100510041005100410051004100510041005},
{1006, 10051006, 100610051006, 1005100610051006, 10061005100610051006, 100510061005100610051006, 1006100510061005100610051006, 10051006100510061005100610051006, 100610051006100510061005100610051006},
{1007, 10061007, 100710061007, 1006100710061007, 10071006100710061007, 100610071006100710061007, 1007100610071006100710061007, 10061007100610071006100710061007, 100710061007100610071006100710061007},
{1008, 10071008, 100810071008, 1007100810071008, 10081007100810071008, 100710081007100810071008, 1008100710081007100810071008, 10071008100710081007100810071008, 100810071008100710081007100810071008}}

Table[((10001 (a + 999) + 1) (10000^n - 1) + 9999 Mod[n, 2] 10000^n)/99999999, {a, 8}, {n, 9}]

你能找出规律来吗？

王守恩

规律来啦！说走咱就走哇！！！你有我有全都有哇！！！

W=FoldList[#2*10^IntegerLength@#+#&,#,PadRight[{},#2-1,{#-1,#}]]&;W[3,10]——谢谢 northwolves！

{3, 23, 323, 2323, 32323, 232323, 3232323, 23232323, 323232323, 2323232323}

W[3,10]——3 表示第 1 个数,  10 表示一共有 10 个数。

W[n, k] ——n 表示第 1 个数,   k  表示一共有  k 个数。