$\sum_{k=1}^448{|sin(x+k)|}/{x+k}>5/2$

mathe

原帖由 gxqcn 于 2008-5-26 20:22 发表
对于 AAx，均有：
|sin(x)| + |sin(x+1)| >= sin(0) + sin(1) = sin(1) = 0.8414...
|sin(x-1)| + |sin(x)| + |sin(x+1)| >= |sin(-1)| + |sin(0)| + |sin(1)| = 2sin(1) = 1.6829...

这是利用 Excel 归纳得到的 ...

这就是我的方法中的一步！！

无心人

加上分母，就不好处理了啊

gxqcn

我也是看了你给的链接，试着“证明”这个不等式的，并试着两个两个的分组，
还是你的每连续三个分组比较好，得到的结果可以更好一点。

gxqcn

原帖由 无心人 于 2008-5-26 21:14 发表


加上分母，就不好处理了啊

可以用 mathe 采用的缩放法，也可用欧拉常数进行估值。

mathe

你整理一下看看.
我在那个帖子里面有些地方写错了，比如放缩以后分母部分应该是
$1/{3+pi}+1/{6+pi}+...$而不是$1/3+1/6+...$

gxqcn

我早看出来了，但论证思想是对的。

无心人

那就是最后得到个定值啊

对了，mathe用什么计算器？
Octave么？
还是dc? bc?
[quote]mathe:
现在用的是Windows的科学计算器，有时再手写一个c程序。如果遇上大数计算，我会先选择bc.遇上矩阵，才会考虑Octave.[/quote]

shshsh_0510

这个应该不难吧
小于Pi的个数为3，故3个一组，分为448/3=148组
由于分母等差增，所以将148组再对折加，形成74组
此时估计应该单调了，再求一下偏导

无心人

Haskell代码
let fun x = sum [ abs(sin(x + y)) / (x + y) |  y <- [ 1.0 .. 448.0 ]
let pi_x = [ x / 1000.0 |  x <- [ 1.0 .. 3142.0 ] ]
let result1 = map fun pi_x

结果似乎是个递减的函数

开头
[4.496998896515244,4.495833927543312,4.494665755125345,4.493494380292428,
4.492319804078475,4.49114202752026,4.48996105165736,4.488776877532164,
4.48759057955119,4.4864060178607525,4.4852182610090425,4.484027310043763,
4.482833166015422,4.4816358299773515,4.4804353029856925,4.479231586099436,
4.478024680380351,4.476816691559683,4.475609841837632,4.474399806435118,
4.47318658641875,4.471970182857948,4.470750596824936,4.469527829394776,
4.468301881645321,4.467072754657195,4.4658437424223125,4.464615179939917,
4.463383441419584,4.462148527946285,4.460910440607757,4.459669180494582
,4.458424748700117,4.457177146320531,4.455926374454768,4.4546771110633525,
4.453427493109591,4.452174708915071,4.450918759582436,4.449659646217136,
4.448397369927428,4.447131931824318,4.44586333302164,4.444591574636009,
4.443322966994674,4.4420530545043135,4.440779985716461,4.439503761750714,
4.438224383729472,4.436941852777866,4.435656170023871,4.434367336598199,
4.433075353634348,4.43178848569356,
结尾
,3.1519677348194826,3.1517631006388105,3.151555358609898,3.1513527727368515,
3.1511477831037094,3.150939687270672,3.15072848581645,3.150514179322549,
3.150296768373367,3.15007625355607,3.1498526354606975,3.1496266126762844,
3.1494067369316574,3.1491837595864305,3.1489576812320026,3.1487285024625984,
3.1484962238752767,3.1482608460699346,3.1480223696492704,3.1477807952188144,
3.147538566500332,3.147301966553141,3.147062270291363,3.1468194783206647,
3.146573591249528,3.1463246096892563,3.1460725342539857,3.145817365560655,
3.145559104229035,3.145302427740661,3.1450507152525553,3.1447959118228495,
3.144538018069834,3.1442770346146474,3.144012962081213,3.1437458010962978,
3.143475552289448,3.1432022162930457,3.1429334316049595,3.1426687280178927,
3.142400938913644,3.1421300649210457,3.1418561066717037,3.141579064800076,
3.141298939943411,3.141015732741752,3.1407294438379605,3.140451824230821,
3.1401770573682626,3.13989921040684,3.1396182839829576,3.1393342787358236,
3.139047195307456,3.1387570343426425,3.138463796489016,3.1381674823969745,
3.1378897210868795,3.137613691440415,3.137334586973524,3.1370524083213853,
3.136767156122005,3.1364788310161456,3.1361874336473954,3.1358929646621183,
3.1355954247094866,3.1353273175882026,3.135057289329324,3.1347841911416645,
3.1345080236519434,3.1342287874896764,3.1339464832871697,3.133661111679522,
3.133372673304621,3.133086543658368,3.132839387458081,3.132589164542282,
3.1323358755127138,3.132079520973916,3.131820101533207,3.131557617800709,
3.131292070389319,3.131023459914732,3.1307738992240313,3.130583998418595,
3.1303910299474986,3.130194994353519,3.1299958921822144,3.129793723981947,
3.1295884903038815,3.129380191701962,3.129168828732935,3.128956845069693]

无心人

所以问题可加强为
证明函数单调递减