analysis with tan n > n
mina_world
There exist infinitely many positive integer n such that (tan n) > n.
-------------------------------------------------------
is this a unsolved problem ?
mina_world
"mina_world" <mina_...@hanmail.net> wrote in message
news:esdrcg$qjk$1...@news2.kornet.net...
Table[N[Tan[n]/n], {n, 1, 100}]
{1.55741, -1.09252, -0.0475155, 0.289455, -0.676103, -0.048501, 0.124493, \
-0.849964, -0.0502573, 0.0648361, -20.541, -0.0529883, 0.035617, 0.517472, \
-0.0570662, 0.0187895, 0.205524, -0.0631841, 0.00797839,
0.111858, -0.072738, \
0.000402348, 0.0690501, -0.088954, -0.00534106, 0.0453367, -0.121248, \
-0.0100511, 0.0305911, -0.213511, -0.0142482, 0.0206564, -2.28221, \
-0.0183382, 0.0135376, 0.215291, -0.0227235, 0.00816604, 0.0926809, \
-0.0279304, 0.00391846, 0.0545569, -0.0348462, 0.00040238, 0.035995, \
-0.0453612, -0.00264952, 0.0250027, -0.0647532, -0.00543801, 0.0177076, \
-0.116409, -0.00813506, 0.0124778, -0.821511, -0.0109156, 0.0085035, \
0.143635, -0.0139962, 0.00533401, 0.0613634, -0.0177018, 0.00269444, \
0.0366853, -0.022616, 0.000402432, 0.0246615, -0.0300012, -0.00167462, \
0.0174566, -0.0433468, -0.00364469, 0.012593, -0.0775273, -0.00560935, \
0.00903615, -0.419072, -0.00768179, 0.0062744, 0.112546, -0.0100123, \
0.00402227,
0.0467542, -0.0128355, 0.00210435, 0.0279852, -0.0165795, 0.000402506, \
0.0189419, -0.0221689, -0.0011713, 0.0135247, -0.0321224, -0.00269126, \
0.00985003, -0.0567848, -0.00423012, 0.00714136, -0.25346, -0.00587214}
Strange, It looks like that tan n > n is impossible.
How do you think about it ?
jank...@hotmail.com
Apparently, yes, although it has been proved that |tan n| > n for
infinitely many n. See
http://www.research.att.com/~njas/sequences/A088306
According to my program, the first values of n that give tan n > n are
1, 260515 and 37362253.
---
J K Haugland
http://home.no.net/zamunda
mina_...@hanmail.net
I showed the next solution about |tan n| > n from you link.
http://www.math.udel.edu/~lazebnik/papers/tan_n.pdf
But...I can't understand it.
Because...
Show that there are infinitely many positive integers n
such that |tan n| > n.
pf)
Lemma) Every irrational real u has infinitely many
Diophantine approximations q=1 (mod 2) with
|u - (p/q)| < 1/(2.q^2).
If n = (2k + 1).(pi/2) + y, where k is an integers and
y is small, say |y| < 1/4,
then
tan n = cot y (Why? if y<0, true. if y>0, tan n = -cot y)
= [1 + (y^2)/2! + O(y^4)] / [y - (y^3)/3! + O(y^5)]
(Strange...maybe, 1 - (y^2)/2! + O(y^4))
= (1/y) + y + O(y^3).(no from my indication.)--(*)
Suppose now that
|y| = |n - (2k+1)(pi/2)| < 1/{a(2k+1)} for some constant a,
(Maybe, if y is very small, I can find "a" by Archimedes.)
then (*) gives
|tan n| > a(2k+1) - O(1) > (2a/pi).n - O(1).
(Why? I can't understand it.)
and I had a question.
If I show that there are infintely many positive integers n
such that |tan n| > b.n for each b >0,
then lim{n->00} (tan n) / n is not exist.
is this possible ?
If possible, How do you show that
there are infintely many positive integers n
such that |tan n| > b.n for each b >0.
David W. Cantrell
> On 4 Mar, 08:12, "mina_world" <mina_wo...@hanmail.net> wrote:
> > Hello sir~
> >
> > There exist infinitely many positive integer n such that (tan n) > n.
> >
> > -------------------------------------------------------
> >
> > is this a unsolved problem ?
>
> Apparently, yes, although it has been proved that |tan n| > n for
> infinitely many n. See
>
> http://www.research.att.com/~njas/sequences/A088306
Interesting. I was unaware of that reference, but now will send a comment
to Neil Sloane linking that sequence to mine below.
> According to my program, the first values of n that give tan n > n are
> 1, 260515 and 37362253.
For increasingly large values of tan(n)/n, see
<http://www.research.att.com/~njas/sequences/A079332>
together with the corresponding MathWorld entry, which shows the results of
further computations by Phil Carmody.
David W. Cantrell
Phil Carmody
> On 4 Mar, 08:12, "mina_world" <mina_wo...@hanmail.net> wrote:
> > Hello sir~
> >
> > There exist infinitely many positive integer n such that (tan n) > n.
> >
>
> Apparently, yes, although it has been proved that |tan n| > n for
> infinitely many n. See
>
> http://www.research.att.com/~njas/sequences/A088306
>
> According to my program, the first values of n that give tan n > n are
> 1, 260515 and 37362253.
Howdy you isometric devil, you ;-)
What do you think of tan(214112296674652)/214112296674652?
FatPhil
--
"Home taping is killing big business profits. We left this side blank
so you can help." -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
A N Niel
> What do you think of tan(214112296674652)/214112296674652?
>
18.0078...
Klueless
> What do you think of tan(214112296674652)/214112296674652?
122925461
534483448
3083975227
214112296674652
118554299812338354516058
1428599129020608582548671
9322105473781932574489648896
1647533557310758242795542778250
19203062276130315764031455655979057
231767240447593988184889934086223330
16132875282857518417293384808417539001172631
378942232820064582240301274646582332037664078
1169809367327212570704813632106852886389036911
22259360648084057667375368818197310731667745986
1531216647248491128766081193927187028939518325390
4719396382295507308889802400540643563335875857787
197186417171362951534971854903816172537236551228132
4368074519151556502674077262299885926704676221372633
100071341106143073658433833323089743969133079989114295
348065656611924979553181377999664160928613441851213716
6909093424425852275983073604700588403624349958919030856
6666170001744260720956920582292251522427910541800763146162
4062825847242558749454381515818934924447734487275988642228022452
146693013073961620081133606483522315887613577021678543650155007535068
329518791919632670629735531264897175374034718291866616652493301652751797778
17828449483728290349398431939095617996113185979221894340290589756033281201121
32991630208838398092602135931203791605001819035062510541111053219832527007486815
66547089287008023293469790243512844929942498779947416679147789941416918656014596762
318517965974027255407526693334056217916400532756769798571090604798499614550103963161
313291567046903023349633815462031424942495539436793088273056692247361092148732658062900
7074622758673691648691237724290739292711386664428478134862212803354906574934753345836141
107776905765690891030082272446082247686659636156848329320136354631864288219685765732595787283007
5396038929117835939477133661542110341912284802753607061567287219193079283699744674488173066659647
8163070518603000766779251550430858310745683336483553497774485060241988494505046246218503375362664014349
24764978478292252438725714300212093709996024921366138837244009822130126788557681607262128273967634934841541
656663628873183465002720935077228312202585023816648634619965107978725953831633144141594852184603577778946951537005011
20096507370463279895781546044277523587973830472902041317878252912888280443194272152276248822107367873121933748229441250962
1570153714759510073736179625727381214761857814223096101904073972970676806568770074074895299243501413409994503856532696600398
36888564043163255092852330431571319785109671719006307374086799238354460814144499604683901407811229531198309873754403649483872
19796965079255068702690376322721856173285004347758066937543207269889681601594134326800984272114465488089723542384041344572622477
...
Phil Carmody
> "Phil Carmody" <thefatphi...@yahoo.co.uk> wrote in message news:878xed6...@nonospaz.fatphil.org...
> > What do you think of tan(214112296674652)/214112296674652?
>
> 122925461
> 534483448
> 3083975227
> 214112296674652
Old ground, well trod. See mathworld. Tanc.