Integral

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New stuff

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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  

Stuff, continued

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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

Haumea resonance

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The resonant angle   in this case is

 

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.

Pi day 2021

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Mathematicians: Let's calculate 50 trillion digits of  .
Astronomers: Let's assume  , but also  .