证明：sinx+sin2x+sin3x+sin4x+sin5x+sin6x＜√22

LeonhardEuler

sinx+sin2x+sin3x+sin4x+sin5x+sin6x<√22

drc2000再来

论坛lu斑主,
他可以将成千上万个三角函数相加,
何况你这里只有区区的六项......

LeonhardEuler

drc2000再来 发表于 2021-11-20 21:31
论坛lu斑主,
他可以将成千上万个三角函数相加,
何况你这里只有区区的六项......

所以怎么证？？？

drc2000再来

我只小学毕业,只会两项相加...

luyuanhong

下面是我在《知乎》上看到的一个解答：

王守恩

sinx+sin2x+sin3x+sin4x+sin5x+sin6x<√22

Table[N[Maximize[{\(\displaystyle\sum_{k=1}^n(\sin(kx))^2, \pi/2≥ x > 0\)}, {x}], 20], {n, 1, 9}]

{{1.0000000000000000000, {x -> 1.5707963267948966192}},
  {3.0982075573105855139, {x -> 0.93592945566132597361}},
  {6.2480381755746526601, {x -> 0.66729107152447274649}},
  {10.448311954990200297, {x -> 0.51861586156240536829}},
  {15.698817762454939368, {x -> 0.42416220549804371282}},
  {21.999493851990074414, {x -> 0.35882892672353959457}},
  {29.350316813205017016, {x -> 0.31094297781381157521}},
  {37.751276177350662178, {x -> 0.27433665956033431348}},
  {47.202366690047109362, {x -> 0.24544357575233972923}}}

王守恩

王守恩 发表于 2021-12-9 09:05
sinx+sin2x+sin3x+sin4x+sin5x+sin6x 0\)}, {x}], 20], {n, 1, 9}]

{{1.0000000000000000000, {x -> 1. ...

sinx+sin2x+sin3x+sin4x+sin5x+sin6x<√22

a(n)=Maximize\(\displaystyle\sum_{k=1}^n\bigg\lceil\sin(kx)\bigg\rceil\)     n=1,2,3,4,....

a(n)=1,2,3,4,4,5,6,7,7,8,9,10,10,11,12,12,13,14,15,15,...
特别地，把重复的数拉出来，有什么规律？
a(n)=4,7,10,12,15,18,20,23,25,28,31,33,36,39,41,44,47,49,52,54,57,60,62,...

denglongshan

https://bbs.emath.ac.cn/forum.php?mod=redirect&tid=18214&goto=lastpost#lastpost
为什么不能发链接