高校数学の質問スレ（医者・東大卒専用） Part438

[過去ﾛｸﾞ] 高校数学の質問スレ（医者・東大卒専用） Part438 (1002ﾚｽ)
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327: １３２人目の素数さん [sage] 2024/11/19(火) 03:57:43.79 ID:V8CHGcRI

To evaluate the limit
lim⁡n→∞(1−1n)n,\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n,limn→∞(1−n1)n,
we can recognize that this expression is related to the definition of the number eee. Specifically, we can rewrite the expression in a more convenient form.
First, we can use the fact that
(1−1n)n=((1−1n)−n)−1.\left(1 - \frac{1}{n}\right)^n = \left(\left(1 - \frac{1}{n}\right)^{-n}\right)^{-1}.(1−n1)n=((1−n1)−n)−1.
Now, we can take the natural logarithm of the expression to simplify the limit:
ln⁡((1−1n)n)=nln⁡(1−1n).\ln\left(\left(1 - \frac{1}{n}\right)^n\right) = n \ln\left(1 - \frac{1}{n}\right).ln((1−n1)n)=nln(1−n1).
Next, we can use the Taylor expansion of ln⁡(1−x)\ln(1 - x)ln(1−x) around x=0x = 0x=0:
ln⁡(1−x)≈−x−x22−x33−…\ln(1 - x) \approx -x - \frac{x^2}{2} - \frac{x^3}{3} - \ldotsln(1−x)≈−x−2x2−3x3−…
For small xxx, we can approximate ln⁡(1−1n)\ln\left(1 - \frac{1}{n}\right)ln(1−n1):
ln⁡(1−1n)≈−1nas n→∞.\ln\left(1 - \frac{1}{n}\right) \approx -\frac{1}{n} \quad \text{as } n \to \infty.ln(1−n1)≈−n1as n→∞.
Thus,
nln⁡(1−1n)≈n(−1n)=−1.n \ln\left(1 - \frac{1}{n}\right) \approx n \left(-\frac{1}{n}\right) = -1.nln(1−n1)≈n(−n1)=−1.
Now, we can take the limit:
lim⁡n→∞nln⁡(1−1n)=−1.\lim_{n \to \infty} n \ln\left(1 - \frac{1}{n}\right) = -1.limn→∞nln(1−n1)=−1.
Exponentiating both sides gives us:
lim⁡n→∞(1−1n)n=e−1=1e.\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = e^{-1} = \frac{1}{e}.limn→∞(1−n1)n=e−1=e1.
Therefore, the final result is:
lim⁡n→∞(1−1n)n=1e.\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^n = \frac{1}{e}.limn→∞(1−n1)n=e1.

http://rio2016.5ch.net/test/read.cgi/math/1723152147/327

350: １３２人目の素数さん [sage] 2024/11/30(土) 17:55:44.99 ID:L4mkprla

f23[n_]:=(
d=IntegerDigits[n];
p=Permutations[d,{5}];
prd=(10 #[[1]]+#[[2]])(100 #[[3]]+10 #[[4]]+#[[5]])& /@p;
i=Position[prd,n];
ans={};
If[i!={},
idx=i[[1]][[1]];
re=p[[idx]];
AppendTo[ans,{10 re[[1]]+re[[2]],100re[[3]]+10re[[4]]+re[[5]]}]
];
ans
)
res=Partition[Flatten@Table[f23[n],{n,10000,99999}],2];

{{51, 201, 10251}, {26, 401, 10426}, {21, 501, 10521}, {15, 705, 10575},

>   {84, 141, 11844}, {60, 201, 12060}, {51, 246, 12546}, {50, 251, 12550},

>   {21, 600, 12600}, {14, 926, 12964}, {41, 323, 13243}, {15, 930, 13950},

>   {35, 401, 14035}, {41, 350, 14350}, {30, 501, 15030}, {51, 300, 15300},

>   {24, 651, 15624}, {75, 231, 17325}, {47, 371, 17437}, {81, 225, 18225},

>   {65, 281, 18265}, {87, 210, 18270}, {21, 906, 19026}, {21, 915, 19215},

>   {86, 251, 21586}, {27, 810, 21870}, {35, 725, 25375}, {47, 542, 25474},

>   {42, 678, 28476}, {32, 926, 29632}, {90, 351, 31590}, {53, 635, 33655},

>   {36, 936, 33696}, {63, 585, 36855}, {80, 473, 37840}, {87, 435, 37845},

>   {65, 641, 41665}, {89, 482, 42898}, {54, 846, 45684}, {65, 704, 45760},

>   {84, 546, 45864}, {57, 834, 47538}, {78, 624, 48672}, {59, 845, 49855},

>   {63, 855, 53865}, {65, 875, 56875}, {68, 926, 62968}, {65, 983, 63895},

>   {72, 936, 67392}, {75, 906, 67950}, {86, 800, 68800}}
{#[[1]],#[[2]],#[[1]]#[[2]]} & /@res

http://rio2016.5ch.net/test/read.cgi/math/1723152147/350

363: １３２人目の素数さん [sage] 2024/12/01(日) 19:45:01.45 ID:URtmhVxU

https://ja.wikipedia.org/wiki/レイランド数

レイランド数（レイランドすう、英: Leyland number）は、数論において次の形で表される数
x^y + y^x
x と y は1より大きい整数
....
また加算の交換性のために x ≥ y の条件は通常レイランド数の重複をさけるために加えられる。(よって 1 < y ≤ x を用いる)

https://oeis.org/A076980に
8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 93312, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169
が例示されている。

問題　各々のレイランド数についてx,yを求めよ。1 < y ≤ x とする。

想定解

{{2, 2, 8}, {3, 2, 17}, {4, 2, 32}, {3, 3, 54}, {5, 2, 57}, {6, 2, 100}, {4, 3, 145},

>   {7, 2, 177}, {8, 2, 320}, {5, 3, 368}, {4, 4, 512}, {9, 2, 593}, {6, 3, 945},

>   {10, 2, 1124}, {5, 4, 1649}, {11, 2, 2169}, {7, 3, 2530}, {12, 2, 4240},

>   {6, 4, 5392}, {5, 5, 6250}, {8, 3, 7073}, {13, 2, 8361}, {14, 2, 16580},

>   {7, 4, 18785}, {9, 3, 20412}, {6, 5, 23401}, {15, 2, 32993}, {10, 3, 60049},

>   {16, 2, 65792}, {8, 4, 69632}, {6, 6, 93312}, {7, 5, 94932}, {17, 2, 131361},

>   {11, 3, 178478}, {18, 2, 262468}, {9, 4, 268705}, {7, 6, 397585}, {8, 5, 423393},

>   {19, 2, 524649}, {12, 3, 533169}}

http://rio2016.5ch.net/test/read.cgi/math/1723152147/363

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