Euler proved that the infinite sum of all the reciprocals of the prime numbers diverges (but extremely slowly).

1/2 + 1/3 is the largest partial sum less than unity. Adding the next term, 1/5, pushes it over 1.

The sum up to 1/271 is just under 2, 1/277 pushes it over.

The sum up to 1/116411 is just less than e, 1/116423 pushes it over.

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