I want to understand this connection better.

Although it sounds the most involved, arctan answers the simplest trigonometric question you could ask. “The full moon subtends seven degrees of my vision; so how big is the moon?” The arctan (with radius = distance from Earth to moon) is the answer.

But what does that simple operation have to do with:

  1. a continuous sum of
  2. an inversion of
  3. one more than
  4. some sequence of numbers
  5. times themselves?

That’s confusing.

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