高中数学题

Question

请问1+1/2+1/3+1/4+....+1/n>=6,求N值

Answer 1

这个数列叫“调和级数”，是发散的，因此没有极限，和>=6,是n=227的时候。
参考计算表：
n = 0
s = 0
A = Table[{i, "--"}, {i, 256}]
While[s < 6, n = n + 1; s = s + 1/n; A[[n]][[2]] = N[s, 6]]
Print[A]

{{1,1.00000},{2,1.50000},{3,1.83333},{4,2.08333},{5,2.28333},{6,2.45000},{7,2.59286},{8,2.71786},{9,2.82897},{10,2.92897},{11,3.01988},{12,3.10321},{13,3.18013},{14,3.25156},{15,3.31823},{16,3.38073},{17,3.43955},{18,3.49511},{19,3.54774},{20,3.59774},{21,3.64536},{22,3.69081},{23,3.73429},{24,3.77596},{25,3.81596},{26,3.85442},{27,3.89146},{28,3.92717},{29,3.96165},{30,3.99499},{31,4.02725},{32,4.05850},{33,4.08880},{34,4.11821},{35,4.14678},{36,4.17456},{37,4.20159},{38,4.22790},{39,4.25354},{40,4.27854},{41,4.30293},{42,4.32674},{43,4.35000},{44,4.37273},{45,4.39495},{46,4.41669},{47,4.43796},{48,4.45880},{49,4.47921},{50,4.49921},{51,4.51881},{52,4.53804},{53,4.55691},{54,4.57543},{55,4.59361},{56,4.61147},{57,4.62901},{58,4.64625},{59,4.66320},{60,4.67987},{61,4.69626},{62,4.71239},{63,4.72827},{64,4.74389},{65,4.75928},{66,4.77443},{67,4.78935},{68,4.80406},{69,4.81855},{70,4.83284},{71,4.84692},{72,4.86081},{73,4.87451},{74,4.88802},{75,4.90136},{76,4.91451},{77,4.92750},{78,4.94032},{79,4.95298},{80,4.96548},{81,4.97782},{82,4.99002},{83,5.00207},{84,5.01397},{85,5.02574},{86,5.03737},{87,5.04886},{88,5.06022},{89,5.07146},{90,5.08257},{91,5.09356},{92,5.10443},{93,5.11518},{94,5.12582},{95,5.13635},{96,5.14676},{97,5.15707},{98,5.16728},{99,5.17738},{100,5.18738},{101,5.19728},{102,5.20708},{103,5.21679},{104,5.22641},{105,5.23593},{106,5.24536},{107,5.25471},{108,5.26397},{109,5.27314},{110,5.28223},{111,5.29124},{112,5.30017},{113,5.30902},{114,5.31779},{115,5.32649},{116,5.33511},{117,5.34366},{118,5.35213},{119,5.36053},{120,5.36887},{121,5.37713},{122,5.38533},{123,5.39346},{124,5.40152},{125,5.40952},{126,5.41746},{127,5.42533},{128,5.43315},{129,5.44090},{130,5.44859},{131,5.45622},{132,5.46380},{133,5.47132},{134,5.47878},{135,5.48619},{136,5.49354},{137,5.50084},{138,5.50809},{139,5.51528},{140,5.52243},{141,5.52952},{142,5.53656},{143,5.54355},{144,5.55050},{145,5.55739},{146,5.56424},{147,5.57105},{148,5.57780},{149,5.58451},{150,5.59118},{151,5.59780},{152,5.60438},{153,5.61092},{154,5.61741},{155,5.62386},{156,5.63027},{157,5.63664},{158,5.64297},{159,5.64926},{160,5.65551},{161,5.66172},{162,5.66790},{163,5.67403},{164,5.68013},{165,5.68619},{166,5.69221},{167,5.69820},{168,5.70415},{169,5.71007},{170,5.71595},{171,5.72180},{172,5.72761},{173,5.73339},{174,5.73914},{175,5.74486},{176,5.75054},{177,5.75619},{178,5.76181},{179,5.76739},{180,5.77295},{181,5.77847},{182,5.78397},{183,5.78943},{184,5.79487},{185,5.80027},{186,5.80565},{187,5.81100},{188,5.81631},{189,5.82161},{190,5.82687},{191,5.83210},{192,5.83731},{193,5.84249},{194,5.84765},{195,5.85278},{196,5.85788},{197,5.86296},{198,5.86801},{199,5.87303},{200,5.87803},{201,5.88301},{202,5.88796},{203,5.89288},{204,5.89778},{205,5.90266},{206,5.90752},{207,5.91235},{208,5.91716},{209,5.92194},{210,5.92670},{211,5.93144},{212,5.93616},{213,5.94085},{214,5.94553},{215,5.95018},{216,5.95481},{217,5.95942},{218,5.96400},{219,5.96857},{220,5.97311},{221,5.97764},{222,5.98214},{223,5.98663},{224,5.99109},{225,5.99554},{226,5.99996},{227,6.00437},{228,--},{229,--},{230,--},{231,--},{232,--},{233,--},{234,--},{235,--},{236,--},{237,--},{238,--},{239,--},{240,--},{241,--},{242,--},{243,--},{244,--},{245,--},{246,--},{247,--},{248,--},{249,--},{250,--},{251,--},{252,--},{253,--},{254,--},{255,--},{256,--}}

Answer 2

解：∵f（x）是定义在R上的奇函数
∴f（0）=
0
且f（﹣x）=
﹣f（x）

（注：现在解决单调性）
令x1、x2∈R，且x1＜x2
（作差）∴f（x2）－
f（x1）=
f（x2）—f【（x1—x2）＋x2】
=f（x2）－f（x1—x2）－f（x2）
=﹣f（x1—x2）

又当x＜0时，f（x）＜0。
（整体法）
又x1、x2∈R，且x1＜x2
∴x1－x2＜0
∴f（x1—x2）＜0
∴f（x2）－
f（x1）=
﹣f（x1—x2）＞0
即：f（x2）＞
f（x1）
∴f（x）在R上增函数

又不等式f（mx²)－2f(x)＞f(m²x)－2f(m)
(m＞0)
∴f（mx²)－f(x)－f（x）＞
f(m²x)－f(m) －f（m）
(m＞0)

又f（x）是定义在R上的奇函数
∴﹣f（x）=
f（﹣x）

∴不等式转化为：f（mx²)＋f(﹣x)＋f(﹣x)
＞
f(m²x)＋f(﹣m)＋f(﹣m)
(m＞0)

∴f（mx²－2x）＞
f（m²x－2m）
（m＞0）

又f（x）是R上的增函数
∴mx²－2x
＞
m²x－2m
（m＞0）

即：mx²－（2＋m²）＋2m
＞ 0
（m＞0）
∴（x－m）（mx—2）＞0
（m＞0）
。。。（1）。。。

①当m=√2时，m
=
2／m
=√2
∴不等式（1）转化为（x－√2）²＞0
∴x≠√2
即：当m=√2时，不等式解为{x|x≠2}

②当0＜m＜√2时，m ＜
2／m ∴不等式解为x＜m或x＞2／m
即；当0＜m＜√2时，不等式的解为（﹣∞，m）∪（2／m，﹢∞）

③当m＞√2时，m ＞
2／m ∴不等式解为x＜2／m或x＞m
即；当m＞√2时，不等式的解为（﹣∞，2／m）∪（m，﹢∞）

综上所述，①当m=√2时，不等式解为{x|x≠2}
②当0＜m＜√2时，不等式的解为（﹣∞，m）∪（2／m，﹢∞）
③当m＞√2时，不等式的解为（﹣∞，2／m）∪（m，﹢∞）

Answer 3

终边在Y轴上的集合是kπ+π/2,终边在直线Y=X上的角的集合是kπ+π/4
度和弧度转化150°30′=150.5*（π/180）
和-6分之π终边相同的角是集合是弧度制2kπ-π/6,角度制是k*180度-30度

Answer 4

kπ+π/2
kπ+π/4
150.5*（π/180）
角度制是k*180度-30度
弧度制2kπ-π/6

Answer 5

n足够大总是可以的
这个是发散的，要多大有多大~