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![{\displaystyle \int _{0}^{\infty }{\frac {\sin {t}}{t}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2dea61bc58f0b6168c8e8285727cb3854cbee48f)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{101}}\right)}}{\left({\frac {t}{101}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f244f1e29167a3e76fe663090710f392086487fb)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{101}}\right)}}{\left({\frac {t}{101}}\right)}}\cdot {\frac {\sin {\left({\frac {t}{201}}\right)}}{\left({\frac {t}{201}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/17226649db6a4f6a13ab3ba4e45f8118370af03d)
![{\displaystyle \vdots \!\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/159bdb83eaa4b80a12eb764ddb8be1e505de69ce)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{101}}\right)}}{\left({\frac {t}{101}}\right)}}\cdot {\frac {\sin {\left({\frac {t}{201}}\right)}}{\left({\frac {t}{201}}\right)}}\cdots {\frac {\sin {\left({\frac {t}{100n+1}}\right)}}{\left({\frac {t}{100n+1}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1336ac48a0d7c97be14a446d7a060fbb30256ab0)
- True for every
![{\displaystyle n<15341178777673149429167740440969249338310889,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/f13a589d1de885820f0bd4191931f0c5467b0714)
- but false for all
larger than that.
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![{\displaystyle \int _{0}^{\infty }{\frac {\sin {t}}{t}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2dea61bc58f0b6168c8e8285727cb3854cbee48f)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{2}}\right)}}{\left({\frac {t}{2}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d589341989fc2cd955451e03b8bb4ef87ec96007)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{2}}\right)}}{\left({\frac {t}{2}}\right)}}\cdot {\frac {\sin {\left({\frac {t}{4}}\right)}}{\left({\frac {t}{4}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3266aff1b5b1d3d4e2a86091f940f5852cb2b68e)
![{\displaystyle \vdots \!\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/159bdb83eaa4b80a12eb764ddb8be1e505de69ce)
![{\displaystyle \int _{0}^{\infty }{{\frac {\sin {t}}{t}}\cdot {\frac {\sin {\left({\frac {t}{2}}\right)}}{\left({\frac {t}{2}}\right)}}\cdot {\frac {\sin {\left({\frac {t}{4}}\right)}}{\left({\frac {t}{4}}\right)}}\cdots {\frac {\sin {\left({\frac {t}{2^{n}}}\right)}}{\left({\frac {t}{2^{n}}}\right)}}}{\text{ dt}}={\frac {\pi }{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5dcd395e44cc4291aa4ffd7150cea170081849f2)
- True for all
![{\displaystyle n}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
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Behind this "magic" is the following: Let
be a sequence of positive real numbers,
Then
if and only if
diverges, but very slowly (it grows logarithmically). The sum is smaller than
until
gets to approximately
In contrast,
converges, and
for all
An example that diverges even slower is
This will not fail until
1±1
5±1
18±1
52±1
149±3
411+45
−41
1100+270
−220
2850+1200
−870
7210+5000
−3000
17900+19000
−9200
43500+66000
−26000
104000+230000
−71000
The resonant angle
in this case is
![{\displaystyle \phi ={\rm {12\cdot \lambda -{\rm {7\cdot \lambda _{\rm {N}}-{\rm {5\cdot \varpi -{\rm {1\cdot \Omega }}}}}}}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/08bf57c751a79c1dcdc8965f94a7b2bc4561e90c)
This angle is librating intermittently, hence why Buie does not classify Haumea as a resonant object.[1] Haumea's ascending node
precesses with a period of about 4.4 million years. It seems like Haumea's 7:12 resonance is broken twice per cycle, once every 2.2 million years, and is reestablished again a few hundred thousand years later. The resonance will next be broken about 250,000 years from now. See here for the results of my orbit simulation.
Haumea and the other objects in the Haumea family occupy a region of the Kuiper belt where multiple resonances (including the 3:5, 4:7, 7:12, 10:17 and 11:19 mean motion resonances) interact, leading to the orbital diffusion of that collision family.[2] While Haumea is in a weak 7:12 resonance, other objects in the Haumea family are known to temporarily occupy some of the other resonances. For instance, (19308) 1996 TO66, the first member of the Haumea family to be discovered, is in an intermittent 11:19 resonance.
Mathematicians: Let's calculate 50 trillion digits of
.
Astronomers: Let's assume
, but also
.