Why adding up every reciprocal of a prime number never ends
In short: Does the sum of 1/p for all primes converge to a finite limit, or does it grow forever? This video explores a beautiful proof that these values diverge by connecting prime factorization to two pillars of calculus.
While there are infinitely many primes, their contribution to a sum decreases as they get larger. However, the Reciprocal Prime Series still diverges, meaning its total value is infinite. The proof relies on the interplay between the harmonic series and the geometric series.
By utilizing the fundamental principle of prime factorization, the video demonstrates why this specific mathematical behavior must occur. It serves as a sophisticated bridge between elementary number theory and core calculus concepts.
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Sources
- Summary by Intellect · source: The Reciprocal Prime Series (this proof should be taught in calculus!)
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